How multi-lingual were people in the Roman Empire? In Europe during the "Dark" Ages?

Recently, I’ve been reading a number of historical books about the Roman Empire’s decline and the early part of the Middle Ages (a.k.a., the “Dark” Ages in Western Europe). While reading these books, one thing that I wondered about was how multi-lingual people were back then. For example, it’s likely that an educated person living in Rome during the late part of the empire would’ve been able to write and speak both Latin (Classical and Vulgar) and Koine Greek. Those living in the eastern Mediterranean would also know some Aramaic. However, I could also see illiterate people throughout the Roman Empire having to be familiar with more than one language. The same is true for Europe during “Dark” Ages when not only do you have the local variants of Vulgar Latin but also various Germanic (and Celtic) languages and dialects added to the linguistic mix.

As you seem to imply in your OP, it would depend on what you mean by “multi lingual”. Educated Romans often knew Greek, at least in the late Republic and early empire, Julius Caesar often sent messages in Greek to his officers in Gaul in order to prevent leakage of information if message was intercepted. Some parts of the Empire, such as the levant and the Near/Middle East have always had a large amount of multilingualism. Persons might not be literate, but out of necessity they would often be able to communicate with various degrees of skill in several languages.

A Roman soldier who spent years on the German or Persian frontier might pick up a rudimentary ability in the language of his enemies.

Indeed, a lot of specific instances of documented language change in the periods you mentioned have been shown to have probably been caused by widespread multilingualism of one sort or another.

For example, John McWhorter wrote an academic article about how certain ways in which English is grammatically simpler than other Germanic languages (besides Afrikaans) seem to have been caused by Old Danish speaking men living in England in the 8th century (I think it was), speaking imperfect English with their English wives, and the wives adjusting their English by throwing in a few Danish constructions, and the kids growing up speaking this modified English as their first language.

“English is the product 
of a Saxon warrior trying to make a date with an Angle barmaid, and
 as such is no more legitimate than any of the other products of that
—H. Beam Piper

A lot of the Roman leaders in Britain in the first century came from Gaul. I suspect that’s because Gaulish speakers had a fairly easy time learning British, and it’s not unreasonable to think that someone like Agricola would have spoken Latin, Gaulish, and British (at least by the end of his campaign).

How you would go about finding evidence for this is another matter. The number of Latin borrowings* in Welsh / Breton / Cornish suggests a lot of language contact between Latin and British. The number of Latin inscriptions from Britain and the ultimate survival of British suggests that there must have been a bilingual society for the entire period 43–410 AD, but it’s much harder to figure out on an individual level.

*Edited to add: borrowings that are early. Borrowings from medieval and Renaissance Latin don’t count.