Any number given that even approaches a resemblance of exactness seems well beyond our grasp. Very rough estimates is likely all we can begin to hope to get out of the most recent scholarship. Trying to pinpoint any such number for the rather narrow timeframe given here — around the First Jewish War 66–73 AD — is likely overambitious, if not hopeless.
We can only make educated guesses for larger timeframes, and those basically boil down to: they were visibly present, as a distinct community, with a possibly more but even less traceable amount of 'more assimilated looking' (or sounding from their names) members, arriving for the large city of Rome at an estimate of a few thousand individuals around the time of inquiry, equating to probably 1–2% of the city's population. With the total number of inhabitants in Rome not exactly known either.
Rome seems to have been a population sink for the Jews living there, threatened by assimilation, illnesses, and since the latter half of the first century also to the emerging Christian competition. Add to that systematic reductions in numbers caused by events like Claudius' expulsion of Jews from Rome… This one event is often dated quite shortly before the inquired timeframe, but inexactly at that, and the event itself is of unknown significance and extent, sometimes even disputed to have happened at all.
Most previous estimates range between 40,000–70,000 individuals in roughly within that timeframe, with the total range of realistic looking estimates even between 6,000 and 100,000. Example:
At the height of the Roman Empire the Jewish population of Rome grew to about 50,000.
— Leslie C. Dunn & Stephen P. Dunn: "The Jewish Community of Rome", Scientific American, Vol. 196, No. 3, 1957, pp.118–132. jstor
But that is most probably a number too high and surely a number given out with too much confidence for its illusory precision.
An earlier attempt now outdated in its assumptions, limited methodology and far more limited source material already sets out to estimate the number significantly lower although it treats it as a lower bound:
There is no more certainty about the number of Jewish inhabitants of Rome than there is about the number of inhabitants of the city in general. That the Jewish community in Rome was already a numerically very considerable one at the time of the Republic is clear from Cicero's statements. In the following years, this number grew considerably as a result of the increasingly unpleasant conditions in the homeland, the numerous conversions of Romans to Judaism and, under Vespasian and Hadrian, as a result of the uprisings in Judea by the prisoners of war brought to Rome. The Jewish legation that came to Rome in 3 BC on the occasion of the dispute over Herod's will was joined by 8,000 Roman Jews, and twenty-two years later Tiberius was able to raise 4,000 Jewish young men fit for arms in Rome. It is therefore hardly an exaggeration to estimate the number of Jews in Rome in the first imperial period at 40,000.
— Hermann Vogelstein & Paul Rieger: "Geschichte der Juden in Rom.
Erster Band. 139 v.Chr.–1420 n.Chr.", Mayer & Müller: Berlin, 1896. (translated from German, worldcat)
The most recent scholarship treats most of these numbers as rather too high, and still laments the deplorable state of data availability or concrete evidence in either way:
A brief glance at recent scholarship reveals why the study of ancient Jewish demography is in such a fairly poor state: this is not because of a lack of interest or the absence of expertise but because good primary data are so hard to come by. […]
This same study then tries to combine the 'popular from earlier approaches' method of inferring numbers — from literary sources including inscriptions — with inferring and extrapolating demographic data from slightly harder evidence, that is burial sites:
Using this type of approach, it is also possible to propose new figures for the size of Rome’s Jewish population in its entirety. In addition to the Villa Torlonia catacombs, there once existed Jewish catacombs at Monteverde and along the Via Appia. There were also Jewish hypogea along the Via Labicana and in the Vigna Cimarra. […]
A total of 7436 burials over a period of some 300 years equals 24.7 tombs per year. With an average Crude Death Rate of 40 per thousand, the Jewish communities that used mentioned cemeteries for burial cannot have exceeded 620 people or 144 families. Again, these figures are truly sobering in light of the size generally assumed for this community. This size—which is based on an analysis of two isolated references in the literary sources—ranges from 10,000 to 60,000. […]
Still, even if we assume that the 600 persons accounted for on the basis of the archaeological evidence represent a mere 10% of the Jewish community that lived in Rome during the period of the first through fourth centuries, we must nonetheless conclude the earlier estimate of up to 60,000 people is in desperate need of revision. After all, it is simply inconceivable that 99% of the archaeological evidence bearing on this community has been lost (the surviving evidence being 600 out of a hypothesized total of 60,000 people, that is 1%). Since the earlier estimates are extrapolations that are purely hypothetical to begin with, I believe that we should dispense with them altogether. The archaeological evidence speaks a clear language: during the period under discussion the Jewish community in ancient Rome did not encompass more than an average of a few thousand souls at best. Its membership did most certainly not run into the tens of thousands, as all previous writers on this issue have suggested. Confirmation also comes from the Jewish funerary inscriptions: to date, some 600 such inscriptions are known. This number stands in strong numerical contrast to the 40,000 early Christian inscriptions discovered thus far. […]
They are in keeping with a general tendency to criticize inflated population estimates based on literary sources exclusively, with new suggestions concerning the total number of Jews in the Roman Empire, as well as with a new calculation with regard to the maximum number of people residing in Rome under Augustus. According to this calculation, this number amounted to around 450,000 rather than to 750,000 or more, as proposed by previous writers on the subject. Once we contrast Jewish with non-Jewish evidence, the conclusion becomes inescapable: less than 2% of the inhabitants of ancient Rome was Jewish.
— Leonard Victor Rutgers: "Reflections on the Demography of the Jewish Community of Ancient Rome", Les cités de l’Italie tardo-antique (IVe–Vie siècle), Collection de l'École française de Rome 369, Roma: École française de Rome, 2006. (PDF)
For another estimate on how many grains of salt to take when using these numbers given, also beware the ample difficulties that surround inferences from such burial sites. Such catacombs were still rare overall and were coming into seemingly common use as 'exclusively Jewish catacombs' rather late. As the very same author listed in his earlier works, like:
— Leonard Victor Rutgers: "The Jews in Late Ancient Rome. Evidence of Cultural Interaction in the Roman Diaspora", Brill: Leiden, Boston, 1995. worldcat
Yet another estimate that again rests mainly on literary evidence:
Cicero stressed the group’s size, cohesion and political clout: ‘You see how great their numbers are, how they stick together, what weight they carry in public meetings.’ He thus depicted the Jews of Rome as an organized constituency engaged in the capital’s civic life, while remaining connected to the Jewish community across the Mediterranean. Similarly, the biographer Suetonius reported that the Jewish community came out en masse for several nights in a row to pray for Julius Caesar after his assassination in 44 BCE. […]
Although precise numbers are impossible to determine, by the turn of the first millennium there were perhaps 40,000 Jews living in Rome: a mix of slaves, freedmen, freeborn immigrants, and native- born citizens with full political rights.
— Ryan R. Abrecht: "An immigrant neighbourhood in ancient Rome", Urban History, Vol 47, No 1, 2019, pp.1–21. doi
A still fairly recent summary of perspectives on the problem would read:
The Jewish community of Rome probably numbered somewhere in the region of 20,000–60,000 in the early first century AD; Solin takes 40,000 as a likely maximum. The estimates are based on two figures in the ancient sources: Josephus says that 8,000 Jews turned out to oppose Archelaus when he came to Rome to claim his father Herod's throne in 4 BC, and Tacitus says that 4,000 Jews descended from freedmen (i.e. male citizens of military age) were conscripted and sent to Sardinia in AD19 (see p. 42). The Jews therefore composed some thing between 2% and 6% of Rome's population up to AD 19. The expulsions by Tiberius and Claudius would have reduced this temporarily, but probably made little long-term difference. It is, however, unlikely that the Jews of Rome would have been self-reproducing, since most of the factors inhibiting the reproduction rate in the city would have applied to them as much as to the rest of the population (see p. 18). Conversions to Judaism could have helped to maintain the size of the community, but they were surely not happening to the same extent as they later did for the Christians; Solin (1983, 616) thinks it unlikely that there were 'many' proselytes. As there is no evidence of whether the number of Jews or the proportion of the population which they formed remained stable, increased or decreased after the Julio-Claudian period," there are no grounds for estimating how much Jewish immigration there was after the early first century. Epitaphs and rabbinic literature show that there was certainly some.
It is also impossible to know what proportion of the Jewish community at Rome held Roman citizenship before 212. Ex-slaves would normally be citizens; other immigrants from the East were more likely to be peregrini. If the two groups intermarried, the extent of citizenship may have diminished rather than increased within the community, since citizenship could only be inherited by the children of marriages between citizens.
Literary sources about Rome from the Jewish perspective are limited. Some rabbinic first impressions of Rome were mentioned at p. 145. Philo makes some observations in the Legatio based on information he must have acquired when he had dealings with the Jews of Rome during his stay there. Josephus says very little about them despite being one of their number after AD70; it is unlikely that he was on very friendly terms with most of them.
The only evidence surviving from Jews themselves which gives any real insight into their life in the city is epigraphic: the epitaphs from the Jewish catacombs and other places of Jewish burial. This material has many shortcomings: the difficulty of differentiating Jewish from non-Jewish records; the lack of any Jewish epitaphs from before the end of the second century AD. Nevertheless, it is informative about some aspects of the life of the community, such as the number of different synagogues and the titles held within them. It provides information about a small number of Jews who are specifically said to be immigrants: from Aquileia, Catania, Achaea, Laodicea, Caesarea in Palestine, Sepphoris, Tiberias, Area Libani, Thabraca in Numidia, Tripolis (which one is uncertain); there is a notable predominance of places in Syria-Palaestina in the list. One epitaph has a curse formula about the 'wrath of God' which is also attested at Acmonia in Asia, suggesting an immigrant from there. Inscriptions also show the various terms of self-identification which were available, such as Ioudaios and Hebraios (see below), as well as the visual symbols by which the Jews identified themselves.
The state saw the Jews as an identifiable group which received special treatment in certain circumstances. The fact that there were three expulsions of Jews from Rome up to the time of Claudius (see p. 41) shows that they were perceived as 'foreign' at least until that date, since expulsions were only practised against groups which were in some sense foreign. However, their treatment as a special case had some advantages too, such as the permission they were given to make their own collections of money for the Temple even while Jerusalem was outside Roman jurisdiction, and the special arrangements made by Augustus to save Jews from having to collect their corn dole on the sabbath.
Jews were likely to be objects of suspicion but, unlike the treatment of the Germans after the Varus disaster, they were not penalized for the behaviour of their co-religionists elsewhere against Roman armies. Whatever the exact motivation of the expulsions, they were related to local causes in Rome rather than events in Judaea, and they were not repeated during AD 66–70 or the subsequent revolts. There is no evidence that reprisals were taken against the Jews of Rome then, or even that they suffered any particular restrictions during the revolts, although like all Jews they subsequently became liable to the Jewish Tax.
— David Noy: "Foreigners at Rome: Citizens and Strangers", Duckworth: London, 2000, pp.257–258. worldcat