This is a part of the FAQ at /r/AskBibleScholars and I'll paste the answer provided by /u/Flubb

I'll ignore the question of the authorship because that entirely depends on who you want to listen to, but the latest monographs indicate that Aramaic, Hebrew, and Greek were actually all reasonably widespread. Ilan's Lexicon shows that Greek names were immensely popular in Judea for both sexes, and that you can't have that kind of penetration of names without the attending penetration of the language (14% of the Judaean population had Greek names). Wise's work on literacy in Roman Judea indicates that roughly 75% of the people who claimed literacy could at least write in Hebrew. Hebrews wrote to each other in Hebrew but in Greek to outsiders (cf Josephus) - there were no Aramaic collections at Masada or Bar Kokhba.

There are two models for literacy, Finlay's Primitivist model - which suggests only 1-2% of the population would be able to afford literacy, and a contending model which shows that up to 30% of the population consists of a sort of 'middle class'. This contending model for literacy would mean that roughly 8-25% of householders could read - so in a village of 1000 people, containing 200 householders, maybe 30 could read and write Hebrew. Roughly 10,000 people in Judaea would be able to speak and write Greek and that limits the ability to only the elite, and disregards any women, children, or non-elites, and Wise follows down a trail of argument that you could end up with almost 50,000 people being literate in Greek in Judaea. Greek literacy was a little less, perhaps 20% of the householders, but they knew it well - not just a signatory level, but to a decent comprehensive level, much better that they generally knew Aramaic. Wise's argument is slightly more complex than I'm making it, but he ends up with a figure of roughly 65% of male householders were literate to some degree, but the caveat is in how you define literacy. If you aggregate everything together (and it's a fair amount of complex extrapolation), perhaps 5-10% of the male population could read books. 16% could sign their name in the appropriate language (which is a another definition of literacy) and there's quite a bit of wiggle room on those numbers and they only refer to written languages, not what you'd speak. Jews were differently literate in languages for different reasons.

Speaking-wise there's no problem with di or even tri-glossic tendencies, and Judaea is a reasonable example. Languages will be used in particular settings - what you speak in the marketplace is not what you'll speak amongst friends and family. Greek and Hebrew would certainly have been superstrate, with Aramaic being substrate.

The whole question is whether all that written literacy (which tended to clump in the upper echelons) would have made its way down to Jesus et al. He could probably speak all 3 languages (and perhaps Latin), whether his followers (which included a large number of people outside the 12 it should be remembered) could read and write is not really answerable. They could have - whether they did is the other question. Whether they bothered to (and it got lost) or it got collected later on is a different question, but there's nothing to stop them having done so. If you want plausibility, then yes, they could have, assuming they lived long enough.

I'll add that at some point Hezser and Harris might be mentioned, and while they're both very important books, they are, at important points, fatally flawed (see Wise on this).