In general, the Mediterranean religions were open to accepting foreign gods. There were two types of acceptance: as a translation of an existing god, and as a foreign cult. A translation was when one civilization had a god that had similar characteristics as a god in a different culture. When someone wanted to refer to the other culture’s god, they would use the name of their own god. For example, the Carthaginians worshipped the god Baal Hammon. Whenever the Romans wanted to refer to the god the Carthaginians worshipped, they referred to him as Saturn because of the similarities between Saturn and Baal Hammon (eating children). Some cultures had stronger similarities than others, which is why we call it the Greco-Roman pantheon, even though the Greeks and Romans had different names for the gods – it was an act of translation. The other type of acceptance was as a foreign cult, which did have specific rules surrounding whether it was accepted or not. Both the Romans and Carthaginians accepted foreign cults, and they acknowledged these gods for worship- but fully understood their foreignness. The Carthaginians began to worship Demeter and Kore in 396 BCE and made sure to use Greek priestesses so they would be worshipped correctly. The Romans were famous for accepting foreign cults, such as Magna Mater and the cult of the Persian Mithras. While I can’t say much about the Celts, the main Mediterranean civilizations were very accepting of other polytheistic religions.

Sources:

Archaic Roman Religion by Dumezil

Foreign cults in Rome by Orlin

Carthage: A History by Lancel

I asked a somewhat similar question last week that /u/Astrogator gave an exceptionally comprehensive answer to.