Composers didn't write their own lyrics! An opera would have had the words written by a special type of poet called a librettist, but in this case (as it's not an opera) Mozart was setting traditional words for a Requiem Mass. I don't think we know who wrote the words for the liturgy but they are very very old, and other composers set them to music. So the words are a Mass, like a prayer, they're not just pretty words. I'm taking a guess you weren't raised Catholic because you asked this question, but maybe you have some experience with Protestantism, and the only Protestant equivalent I can think of is maybe all the different settings of the Doxology if you know that? So in some churches it's sung to Duke Street, in others to Old 100th, and many other songs, but it's the same prayer, the tunes are just for style.

So a bit more general information about Mozart and his librettists for opera, which is probably closer to what you're actually asking about. Here's how you made an opera in the early Classical period: so someone who wanted an opera and would commission it, and they'd probably hire the poet first because the poetry was seen as more important than the music (though this is starting to evolve in that period). So the poet would write the libretto for an opera first, and then the composer was brought in to set the words. Sometimes this would happen years apart. Sometimes when the composer was writing the music he'd kick it back to the librettist and say such and such wasn't working and ask for edits, or just cut some of it out himself maybe, if the librettist was long gone. Or you could get a second librettist to edit the first one's work. But music was (allegedly!) the servant to poetry, so the poetry came first, and it was set by a proper poet.

Mozart was a more involved composer with his librettists than others, and towards the end would work with them while they were writing the poetry, as opposed to after. One of his most productive relationships was with a man named Lorenzo Da Ponte (who had a CRAZY life and came to live in America in the 19th century and founded the Italian department at Columbia, look him up), who was the librettist for three of Mozart's most beloved operas: Don Giovanni, The Marriage of Figaro and Così fan tutte.

Theoretically though... the composers (and the singers) would have been pretty proficient in the languages they were working with. Formal Italian and Latin would have been part of musical conservatory training for the singers and composers who went through that, conservatory training in the 18th c. was really pretty academically robust, and turned out some highly intelligent musicians. But as Mozart was a Boy Genius and didn't go to a conservatory, I would assume his father taught him his Latin and other languages he worked with.

Some, but not all, musicians learned Latin. For some, their positions included teaching Latin as part of their duties (J.S. Bach comes to mind), so I guess we can assume it was not incredibly rare to find some musicians capable of doing that (they were permitted to have somebody else do the teaching, too). Learning Latin was a requirement for academic education, and some musicians had contact with that kind of path. Some music treatises from the 17th-18th century were still in Latin, and pretty much all from before that were written in Latin (but I guess it's safe to assume not every musician would need to be in contact with those). Some musicians were ordained in the Catholic Church (like Vivaldi), so their education would include Latin.

If they were composing religious music with a Latin text, they would certainly be using the approved versions (already mentioned by Nebuchadnezzar-II). It would have been a very bad thing to present a composition with the wrong text. Imagine you are a graphic designer today and you deliver work with obvious dumb mistakes, you are not going to be in business for long with that kind of thing.

As /u/caffarelli mentioned, musicians would get somebody else to write the libretto, or would set some poet's work to music. Some musicians were able to speak/write more than one language. In many cases it was useful to be able to do that.

Musicians did travel back then, they would aspire to get prestigious positions and study under famous masters... If they were dealing with operas in Italian, it would not be rare to find them learning Italian (and they would probably interact with Italian singers, because there were Italian superstars in that world).

I don't know of any famous mistake in lyrics. I guess such a thing could happen, but that kind of mistake would be eventually fixed (preferably VERY soon). Music was not a hobby for these people, they were not having fun with their mates. They were aiming for the stars, trying to produce successful operas, trying to woo everybody with their new requiem commissioned by a rich family or religious organization... They had many reasons to get things right, and would try to fix mistakes quickly. I hope somebody can find an example of such an incident.

There were some mistakes when copying music, or when printing it. And we find errata.

As to where composers learnt their Latin the answer is relatively simple. They learnt it in school and it was required knowledge for any gentleman of learning. From what I remember from "Music in the Castle of Heaven" by Gardiner Latin was one of the few things taught in Bach's school and his marks were on the whole very bad. Yet several of the pieces of the Musical Offering bear Latin subtitles At the other end of the timescale Vaughan Williams attended Charterhouse school where Latin is still a compulsory part of the curriculum to this day. As for Mozart himself I can only assume that he learnt from his father on one of his gargantuan tours around Europe, although I think it is safe to say that his Latin was probably quite bad, as was his written German.