Curious to know what languages would've been spoken and used in correspondences during this time, specifically in Transylvania.
Roumania consists mostly of the three main sections Transylvania, Wallachia, and Moldavia.
In the 11th century (1001 to 1100) and the 12th century (1101-1200) Transylvania, Wallachia, and Moldavia were usually part of separate realms.
Wallachia, for example, was ruled by Turkic peoples such as the Pechnegs in most of the 11th century and the Cumans in the 12th century. Thus one or more Turkic languages would have been spoken by some of the people in Wallachia in the 11th and 12th centuries.
The Magyars conquered much of Central Europe at the end of the 9th century. According to Gesta Hungarorum, the Vlach voivode Gelou ruled Transylvania before the Hungarians arrived. The Kingdom of Hungary established partial control over Transylvania in 1003, when king Stephen I, according to legend, defeated the prince named Gyula. Some historians assert Transylvania was settled by Hungarians in several stages between the 10th and 13th centuries, while others claim that it was already settled, since the earliest Hungarian artifacts found in the region are dated to the first half of the 10th century.
So the Magyars in Transylvania spoke Hungarian in the 11th century, and quite possibly ruled over subject peoples there who spoke various Slavic languages and dialects.
The historian Kurt Horedt dates the entering of the Hungarians in Transylvania in the period between the 10th century and the 13th century. In his theory, the Hungarians conquered Transylvania in five stages: 1st stage – around year 900, until Someșul Mic river
2nd stage – around year 1000, Someșul Mic valley and the middle and lower course of Mureș river
3rd stage – around year 1100, until Târnava Mare river
4th stage – around year 1150, until the Olt River line
5th stage – around year 1200, until the Carpathian Mountains
Conflicting theories exist concerning whether or not the Romanized Dacian population (the ancestors of the Romanians) remained in Transylvania after the withdrawal of the Romans (and whether or not Romanians were in Transylvania during the Migration Period, particularly during the Magyar migration). These theories are often used to back competing claims by Hungarian and Romanian nationalists.
Several theories address the issue of the origin of the Romanians. The Romanian language descends from the Vulgar Latin dialects spoken in the Roman provinces north of the "Jireček Line" (a proposed notional line separating the predominantly Latin-speaking territories from the Greek-speaking lands in Southeastern Europe) in Late Antiquity. The theory of Daco-Roman continuity argues that the Romanians are mainly descended from the Daco-Romans, a people developing through the cohabitation of the native Dacians and the Roman colonists in the province of Dacia Traiana (primarily in present-day Romania) north of the river Danube. The competing immigrationist theory states that the Romanians' ethnogenesis commenced in the provinces south of the river with Romanized local populations (known as Vlachs in the Middle Ages) spreading through mountain refuges, both south to Greece and north through the Carpathian Mountains. Other theories state that the Romanized local populations were present over a wide area on both sides of the Danube and the river itself did not constitute an obstacle to permanent exchanges in both directions; according to the "admigration" theory, migrations from the Balkan Peninsula to the lands north of the Danube contributed to the survival of the Romance-speaking population in these territories.
Political motivations—the Transylvanian Romanians' efforts to achieve their emancipation, Austro-Hungarian and Romanian expansionism, and Hungarian irredentism—influenced the development of the theories, and "national passions" still color the debates. In 2013, authors of The Cambridge History of the Romance Languages came to the conclusion that the "historical, archaeological and linguistic data available do not seem adequate to give a definitive answer" in the debate. Their view was accepted by scholars contributing to The Oxford Guide to the Romance Languages, published in 2016, which concludes that "the location and extent of the territory where "Daco-Romance" originated" is uncertain.
Thus the scientific and historic evidence as to what proportion of the population might have spoken Vlach languages and dialects ancestral to modern Romanian in Romania in general and in Transylvania in particular during the 11th century is not very clear or unambiguous. Instead the available evidence is ambiguous and uncertain enough for nationalistic Hungarian and Romanian historians to have quite contradictory interpretations.
what languages would've been spoken and used in correspondences during this time, specifically in Transylvania
The two main points
11th century in Hungary and Romanian principalities
written (political) correspondence/documents
do not fit together very well.
There are only a few candidates for written documents:
Hungarians had an old script but its use and documentation is scarce. Most probably few could use it anyway. It was probably not intended for "correspondence". Christianity was adopted by King Stephen at the very beginning of the 11th century. After that the language used in official documents is Latin for internal needs and relations with the Pope and the Catholic powers, but possibly Greek might have been used in relation with the Byzantines.
Romanian and Slavic populations have not left any written records from the 11th century
German colonization has started a century later, just like the establishment of the Teutonic Knights
Petcheneg, Cuman and other Turkik warriors were illiterate. The fact that there is a Codex Cumanus made by missionaries is not a proof against this. Roman Catholic Diocese of Cumania is a 13th century fact.
If by "correspondence" other means than written ones are implied, various spoken idioms can be inferred corresponding to any ethnic element noted there and then, but with little to no epistemic content.