It depends, but generally it's the former. The Greeks and Romans did not really use punctuation, and they didn't put spaces between words. Reading was almost always done aloud, and their languages were much more phonetic than English, so the text would make sense once you've sounded it out. Obviously there was still room for ambiguity and you might have to read something a couple times before you got it, but that's just how it was done.

However, there are plenty of texts for which the oldest surviving copy is a medieval manuscript. Many of these do have punctuation, but they often serve a more rhetorical function--that is, they indicate where to pause when speaking, rather than dividing up clauses and sentences.

But when it comes to translating, yes, the translator basically has to make up his own punctuation. Because of grammatical differences, it's often necessary to reorder, cut up, or recombine phrases and sentences so that they actually make sense in English. In these cases, the new punctuation may have little or no relation to the original structure of the sentence.