Do either of them pre-emptively charge into the homes of people they don't even know are criminals, attack them, and take their stuff?
No, they usually know they are criminals. They just don't usually wait around for silly things like search warrants.
And as said, you're stealing back your stuff. There is no jurisdictional nonsense involved. No necessity to have warrants.
You should also probably know that "hero" is an extremely subjective term. It's not like our characters didn't do the same exact things in City of Heroes. They did. A LOT.