Depends completely on the universe in which it is set. If keeping your real identity a secret is important then you'll want a mask. If there's a reason for flashy costumes in that world then you might also find a reason to wear a mask.
You have to make allowances for their reality, those domino masks worn in many comics will not hide an identity too well for some high profile individuals - when we read these comics we share in that world's reality that it is enough though.
I've honestly been good with both versions - if it makes sense and doesn't clutter up the story or appearance of the characters then it usually doesn't matter to me. Certainly a story can be made on examining the whole mask thing - just as the 'No Capes' was deconstructed in that great movie The Incredibles