As a left-hander, it's very obvious to me. That said—because of the above point—my right hand is much stronger and more adept at playing. In short, I'm right handed when playing the piano.
No sarcastic comments please, I well know more practice and playing those damn Czerny scales ad nauseam would have restored proper balance. :-)
The specialization goes both ways, though, I'm much better playing Alberti bass with my left hand compared to the right.
I've never had any pretense at being good enough to entertain people with those works as everyone knows them so well (from professional recordings). Even Mozart's a problem here. For example the Romance in the D Minor Concerto, K.466 looks deceptively simple (at least in parts) but it's anything but after hearing someone like Brendel play it. Everyone knows it so well it's not worth the embarrassment of even trying (except perhaps in secret).
(Mozart has a habit of looking simple until one tries to play it, Bach is none of that—one knows what one's in for at pretty much first glance.)