[Agema]

There are hints looking at language development. For instance, "c" seems to have developed from "k" in much earlier Latin, which makes it very likely that it was pronounced hard, and there are other hints. You might be able to tell from poetry - inaccurate prounciation can wreck meter and rhyme.

Also I think that most of the population - even Italians in classical Rome - would not generally have spoken what we know as classical Latin, which was really a very formal and proper form used in literature and by the upper classes. Most people would have spoken Vulgar Latin, which would have varied greatly across nations and peoples in the Roman empire with some differences in pronunciation and vocabulary. You could compare with different uses of English worldwide, although the comparison fails at the level that English does not really have a formal standard in the way (to the best of my knowledge) that Latin did.