Wearing a mask means you are in your hero persona. Not wearing a mask means you're in your "secret identity" personal, assuming your character has one. That said, changing the whole costume is recommended to avoid confusion.

As for which identity to use, if you have two personas that each have a reason to be there at start of game, take your pick, just be ready for a quick change if someone says "Gee, I really wish I could talk to $other_character right now...". =:)