In terms of specific individuals, the Sramana gymnosophist known as Zarmarus or Zarmanochegas is usually identified as a Buddhist (though this is not definite - part of a greater confusion over whether the Sramana mentioned in classical sources referred to Buddhists or not).
He was dispatched by an Indian ruler to meet with Augustus, and killed himself by immolation in Athens. This embassy is well attested in the works of ancient writers, such as Plutarch, Strabo, and Cassius Dio. Strabo wrote, for instance, that:
His naked body was anointed and wearing only a loin-cloth he leaped upon the lighted pyre with a laugh. The following words were inscribed on his tomb: "There lies Zarmanochegas, an Indian from Bargaza, who immolated himself in accordance with the ancestral customs of the Indians."
- McLaughlin, Raoul. Rome and the Distant East: Trade Routes to the Ancient Lands of Arabia, India and China. A&C Black, 2010.
Much more tenuously, the Mahavamsa (a Buddhist history of Ceylon) records a delegation of Buddhist monks from "Alexandria". They were led by the Greek elder Mahadhammarakkhita, and arrived at Ceylon to attended the erection of the Mahathupa at Ruanwellil.
Mahadhammarakkhita, thera of Yona, accompanied by thirty thousand bhikkhus from the vicinity of A'lasadda, the capital of the Yona country, attended.
It is agreed that Yona meant Greek and that A'lasadda referred to Alexandria, but it isn't altogether clear which Alexandria. It seems scholars generally think it refers to the Alexandria of Bactria or the Alexandria of Arachosia. Alternatively, and I don't think anyone advocates this except the dubious Arthur Lillie, but it might also possibly be the Alexandria of Ptolemaic Egypt.
More generally speaking, the idea that there were Buddhists in the Roman Empire is not very exceptional, despite the overly defensive objections of some.
It is well established that there was at least some interaction between the Roman Empire and the Indian subcontinent, at a time when Buddhism was ascendant in the latter. Part of this interaction was the movement of individuals, such as merchants and diplomatic envoys, between the two regions.
[T]he fact that Roman subjects constantly visited India but Indians seldom visited the Roman Empire (except Alexandria and Asia Minor) is reflected in such evidence which we can collect.
- Warmington, Eric Herbert. The Commerce Between the Roman Empire and India. Cambridge University Press, 2014.
Indeed, an Indian presence was attested to in Alexandria, a major cultural and commercial crossroad. The orator Dio Chrysostom, for instance, remarked that:
Ethiopians and Arabians from distant regions and Bactrians, Scythians, Persiand and a few Indians all help to make up the audience in your theatre and sit beside you on each occasion.
Probably some of the Indians went on to Rome, though the evidence is lacking. As a side note, Indian was one of the nationalities the poet Martial mentioned in his lampooning of a certain (fictitious) Caelia:
For you from his Egyptian city comes sailing the gallant of Memphis, and the black Indian from the Red Sea ... What is your reason that, although you are a Roman girl, no Roman lewdness has attraction for you?
To be sure, Indians were certainly not a common sight in much of the Mediterranean. At the same time, the evidence showed that they were present in Alexandria. In all likelihood some visited some other parts of the Roman Empire. It is not a stretch to conclude that some of those Indians were Buddhists.
Separately, Ashoka the Great left records of sending Buddhist missionaries into the West, including to Alexandria. They don't seem to have had much impact, though we know diplomatic (and commercial) contact continued thence.
