In my experience, "We're doing naive set theory" means something like, "We'll assume, without further justification, that no Russell-style paradox applies to any predicate P where we will actually want to write {x : P(x)}. We'll just assume the existence of a set answering to this description for any P that we need. We know that there are predicates for which this is not allowed, but we'll just hope that everything works out okay in the cases where we do it."

The phrase "naive set theory" also connotes a certain cavalierness about whether the elements in one's sets are themselves constructed out of sets (as in ZF) or whether instead one is working with urelements (objects in sets that are not themselves sets).