It didn’t change. After they returned from Babylonian Exile, the Jews typically assimilated the language of whatever country they were in (starting with Aramaic) while preserving Hebrew as a language of worship and religious scholarship. (Like Church Latin in the Middle Ages.) E.g., the large Jewish population of Alexandria in Egypt spoke Greek; the OT was translated into Greek for their use. (http://en.wikipedia.org/wiki/Septuagint) In went on like that – from the Middle Ages to the 20th Century, Sephardic Jews (Spain, Italy, Ottoman Empire) spoke a Romance language, Ladino, while Ashkenazi Jews (Eastern Europe) spoke Yiddish (a German dialect with Hebrew loanwords). By the time Jews started colonizing Palestine in the early 20th Century, Hebrew was not a living, spoken language anywhere and had not been for well over 2,000 years; but all religious Jews had some knowledge of it, and the Zionists decided to revive it as their national language. It was the only language all Diaspora Jews had in common. Modern Hebrew has little in common with Aramaic but a lot in common with ancient Hebrew. (In fact it’s almost identical, except they had to add a lot of new vocabulary for things that simply didn’t exist in ancient times. http://en.wikipedia.org/wiki/Hebrew_language)
It would be Aramaic and Greek, not Aramaic and Latin. Greek was the lingua franca of the east and the language of administration in the eastern Roman Empire.