We have really limited sources when it comes to the oral traditions of ancient Greek mythology, but I'm going to focus on two things--how easy it would be to travel between city-states and thus share the myths, and what we know about the myths now.

A lot of what you were probably taught in any class that touched on mythology is going to come from Hesiod's Theogony, where he lays out the names of the gods, the creation myth, and so forth. The Theogony is really as early as it goes when it comes to written myth. Hesiod is thought to have been active around the same time as Homer. The translations of Hesiod that I have read don't get region-specific as far as their epithets of gods, so there's not much luck there, but I mention it because first of all: Hesiod was highly regarded by the Greeks, so at least the creation myth was pretty ubiquitous and secondly: he mentions cities and rivers in places where he didn't live, so he was hardly isolated. Therefore, at least by around 700 BC, certain stories were shared.

When it comes to the different versions of myths you've read (which might be why you ask the question?), a lot of those end up being Roman-related, or the fault of later authors who were writing about the mythology. Ovid, for example, drew heavily on a lot of myth sources for his Metamorphoses and ended up changing the 'canon' (for lack of a better term). This is a really minor example, but as an example of some of the differences: Ovid says that Persephone ended up staying in the Underworld for half of every year. The Bibliotheca, a mythology book said to have been written by Apollodorus of Athens, says she only stays there three months. And so, depending on what version you're drawn on, she either eats only one pomegranate seed and that's enough to damn her, or she eats a number of pomegranate seeds equivalent to the number of months she has to stay, and so on.

Back to the ancient Greeks and the oral traditions. The Athenians, at least, tended to be pretty stingy about who got citizenship in ancient Athens. You might imagine that would keep them isolated but that's not the case--by at least 500 BC, we know that foreigners could live in Athens, although not as full citizens. This is of course at least a whole two centuries after Hesiod and Homer, but it's really difficult to be precise that far back. By that point, the people of each city-state were pretty effectively mixing. Each city-state might hold a god to be more important--Apollo was going to get better traffic in Delphi than in Athens, for instance--but the oral myths would be shared and by that point things were starting to get written down anyway.

Basically--while they may have been isolated before, at least by the time that myths were getting written down, they were relatively ubiquitous. A lot of the myth variations you may know will come from later authors. 'Immigration laws' among city-states started to get relaxed around 500 BC and so certainly by that point, the myths would be shared, although different city-states still had certain gods they held as more important to them.