(My answer to this question at Quora.com)
The experts are divided on this question: some think that Judeans carried on speaking Aramaic after their return from Babylon, and used Hebrew only in prayer and Scripture writing and Talmudic debates; others think that they reverted to speaking Hebrew on return from Babylon (or had even maintained it while in Babylonian exile), and spoke Aramaic as a second language, since it was the lingua franca of the entire Middle East.
My sense, based on what I know about the Hebrew script and the dynamics of language and patriotism elsewhere and in other periods of history, is that it was largely a social thing.
Unlike the Assyrians before them, the Babylonians had not exiled the entire Judean population when they conquered it, but only the royal family and upper (educated) classes. After 75 years in exile in Babylon, the returning Judeans saw themselves, by dint of their pedigree and their life in the imperial capital, vastly superior to the ‘poor of the land’ (II Kings 24:14) who had stayed behind, and regarded their knowledge of Aramaic (and use of Aramaic script) as a badge of their social and educational status. Their attitude towards Hebrew was therefore highly ambivalent: if it weren’t for the undeniable fact that it was the language of Mosaic law and the Davidic dynasty, they would have happily dropped it entirely.
However, this attitude likely faded over time—much as French gradually gave way to a Francified version of Old English among the Plantagenets and Anglo-Norman aristocracy in England in the 12th–15th centuries, as they gradually began to regard themselves as English rather than Norman or French.
This trend likely accelerated after Alexander the Great conquered the entire Middle East, and Greek replaced Aramaic as the official imperial language. By Hasmonean times (the 2nd century BCE), the swelling patriotism against the decidedly foreign Hellenist regime, and the desire to return to the heroic independent Israelite past undoubtedly led to a resurgence of Hebrew—as evident from the coins of that period, which pointedly used the original Hebrew script, rather than the Aramaic one that the Talmudic Sages preferred:
From that point on, Judeans likely carried on with a kind of uneasy bi- or trilingualism: patriotic, religious Judean leadership in the countryside speaking Hebrew but using Aramaic in Talmudic studies; Hellenized Judeans in Jerusalem and other large cities preferring to speak Greek, but retaining knowledge of Hebrew as a nod towards their national heritage, and only reluctantly acknowledging Aramaic; and the toiling masses speaking largely Hebrew, with a working knowledge of Aramaic and/or Greek for dealings with officialdom—rather like their counterparts in the Holy Roman Empire a thousand years later, who spoke Old French or Old German at home, but encountered Latin at church or in official documents.