Not necessarily. It's very possible for one system to embed inside the other. In this case you can often talk about exactly the same axiom/theorem/whatever within different inferential systems.

An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic. That said, we can use flattening to embed classical logic inside of intuitionistic logic and then recover a whole family of propositions of which AC is one and also see why it has a difficult time holding in IPL.