(Inspired by this comment).
Does halacha permit Jews to speak languages other than Hebrew? Bedieved? Lechatchilah?
Evidence against there being such a prohibition includes:
Speaking (and writing) other languages has been widespread practice for more than two millennia. While it's hard to prove a negative, I've so far never heard of an objection to this.
Some prayers were specifically written in Aramaic, the language of the people, rather than Hebrew.
As pointed out by @minhag, the talmud is mostly written in Aramaic, Rambam wrote in Arabic, and Yeminite Jews recite Onkelos as part of their t'filah. Also, Rashi was clearly fluent in French.
How would Yiddish and Ladino have gotten off the ground if they were forbidden? These languages were formed by German- and Spanish-speaking Jews, respectively.
No source, but I have heard of people objecting to using Hebrew for purposes other than prayer, which would seem to be a problem if the halacha required Hebrew.
We went for something close to two-thousand years with Hebrew not being a living language, yet there were Jews all that time.