A lot of myths do have regional origins -- not that that means people in other parts of the Greek world wouldn't have known them, but they did 'belong' to particular places. If you browse through something like Pausanias' Guide to Greece you'll see an awful lot of examples of local myths, commemorated by local people. And Pausanias is just the tip of the iceberg.

For something more systematic, modern scholarly encyclopaedias will give documentation, sorted by the mythological figures involved. And by encyclopaedia I don't mean the Penguin guide to Greek myth or whatever, I mean things like Roscher's Ausführliches Lexikon der griechischen und römischen Mythologie, or Smith's Dictionary of Greek and Roman biography and mythology, or Farnell's Cults of the Greek states: enormous compilations to satisfy an enormous amount of curiosity. (And they're on the Internet Archive for free! Yay copyright expiration. Though why aren't there books of this kind less than 90 years old? Not sure, but I'd guess lack of demand has something to do with it: who'd buy them, other than a few dozen university libraries?)

Having said that, there was a lot of pan-hellenising and systematising that went on. This is one of the reason that things like the Olympia and Delia festivals were such a big deal, and the Hesiodic poems and the Homeric Iliad: they took bits and pieces from all over the Greek world and synthesised them into an idea of pan-Hellenic identity. Anyone 'Greek' could take part at the Olympia; people came from as far afield as the colonies in southern Ukraine for the Delia. The Iliad puts Aeolians, Achaians, and Ionians from all over the Greek mainland in a single myth; the story of Jason and the Argonauts did something similar. The Theogony takes a creation myth and combines it with lots of small-time, regional divinities, under the umbrella of a story structure swiped from overseas (the succession myth, taken from the Mesopotamians and the Phoenicians).

In later times, the ancient Greeks had their own myth encyclopaedias too. There was a whole literary-scholarly genre of mythography, starting with figures like Palaephatus in classical Athens and moving on to handbooks like the Library of pseudo-Apollodorus and the Fabulae of pseudo-Hyginus.

So ultimately it's a bit of both. Myths typically had their origin as something regional; but there were certain stories, and certain literary genres, that combined them into something pan-Hellenic.