Roman/Greek religion crossover

[magnificentbeast]

I understand that the roman gods all have their greek counterparts, but I've also read instances of the romans worshipping the greek forms, such as Apollo or Dionysus. I'm confused as to what degree they were considered separate entities. For example, could a Roman settler living in 1st century Greece have been a priest of Apollo, or worshipped both Demeter and Ceres as separate entities? I'm hopelessly confused on their worldview in this matter. Or is this just a matter of which language they might have been speaking in at the time?