You ask for Rome, I answer with Greek (but perhaps it will be a little bit helpful).

So the Greek word for deaf - κωφός (kophos) - also means mute and unintelligent. In an oral culture where literacy rates are low, hearing, speaking, and intelligence are tightly associated. There aren't very many people who are explicitly described as being deaf, but there are a few. For example, Herodotus talks about one of the sons of the king Croesus:

ἦσαν δὲ τῷ Κροίσῳ δύο παῖδες, τῶν οὕτερος μὲν διέφθαρτο, ἦν γὰρ δὴ κωφός, ὁ δὲ ἕτερος...

For Croesus had two sons, one of whom was destroyed, for he was deaf, but the other...

That's a pretty emphatic verb and later on when the king is explaining some of the things he has done to the not deaf son, he says:

εἷς γὰρ μοι μοῦνος τυγχάνεις ἐὼν παῖς: τὸν γὰρ δὴ ἕτερον διεφθαρμένον τὴν ἀκοὴν οὐκ εἶναί μοι λογίζομαι.

You are my one and only son: for I do not consider the other whose hearing is destroyed to be mine.

Which is... pretty harsh. Maybe a little considerable since it would probably be difficulty to be a deaf king at this time, but it shows you a little bit about how sever the ancients saw this disability as (good news for anonymous Lydian prince though, he recovers his ability to speak the day his father dies).

Now this is pretty much our only explicit source on deafness in ancient Greece and while some scholars have argued for a sign language at this period, there just isn't enough evidence to say either way. Certainly it was not a widespread or standardized phenomenon, but it is possible that there were some ad hoc sign language systems used on a local scale.

To turn to another piece of evidence - this time a Greek language source from the Roman empire - there is an evidence for some kind of signing system in the gospel of Luke. When Gabriel makes Zechariah mute (Luke 1:20), the word used is again kophos and is described in a way that indicates he is both deaf and mute. When Elizabeth tells her neighbors that she's going to name her baby John, they go to check with dad. The Greek uses the verb ἐνένευον which means to nod, Zechariah gets the message that they're asking him what to call the baby and asks for a tablet to write it down on.

While it's a lot of work for one little word to do, a number of scholars have taken this to mean that there was some kind of rudimentary sign language. Some have taken it to mean that there was a developed system, but it's more likely that this was a more localized phenomenon. You see similar descriptions of communications by gesture or nod when Greeks/Romans run into peoples with whom they share no mutual language.