Any answer must include the caveat that there were many provinces in the Empire, and within each province there were tens of thousands of different points of view.

Ronald Hutton handles this subject in Roman Britain with his The Pagan Religions of Ancient Britain (1991). We have to put this in the context, understand that the Romans were consolidating power militarily, politically, and with regard to commerce. The fact that they sought out local equivalents of the Roman belief system was actually generous on some level. To a certain extent, Romans were extending an assumption about the supernatural, namely that there were diverse supernatural forces out there, and although there were local entities unique to specific locations, the more powerful forces - the ones we refer to as gods - were likely ubiquitous, and so it would come as no surprise to find locals worshiping Roman gods with local names.

By allowing for the continued obedience of local ritual and belief to indigenous supernatural beings, the Romans were exercising a certain amount of open mindedness. Romans could understand local belief systems because they were cast in "hyphenated terms." Locals could continue their practices, and with or without acknowledgement of the "hyphen." In general, the system worked as long as the locals were not cranky about the Roman overlay.

It's true that Romans were a little fussy about human sacrifice, so there were limits to Roman flexibility. And it is true that during the period of military conquest and political consolidation, the Romans were hard on the local priesthood/druids. But in general, Roman "management" of indigenous belief systems was one of the least obnoxious aspects of Roman rule - at least as far as Britain was concerned.