To clarify, if a roman (for example) was in say in the British Isles, and heard of a local high deity, would he likely think: "Oh, the leader God, we call him Jupiter." or would he think - "Oh, your deity is not a leader, our deity Jupiter is."
Were they competing, or was there a coincidental overlap to any degree of a subset of deities that went by culturally specific names.
There was definitely an overlap, and it was at times forced. For probably the most elaborate example, consider the correspondence between Greek and Roman gods: Zeus was equated to Jupiter, Hera to Juno, Ares to Mars, and so on. This overlap was generally as good as they could make it, but it wasn't perfect; for example, the Romans associated Mars with farming as well as war, which the Greeks had never done with Ares. Though I haven't found any sources, I've heard that the Romans periodically attempted to align the God of the Jews with some member of their pantheon but were never able to agree on whom.
The Romans weren't the first people to do this, either; as early as Herodotus, the Greeks considered the oracle of Amun in the Egyptian desert to be an oracle of Zeus. On his visit there, Alexander even worshiped "Zeus-Amun."