I'd say "private property" trumps "right to carry". I'll still shop there, and if I'm armed, I'll make sure nobody knows it. If they catch me, and ask me to leave, I will. I can't imagine they expend any effort on spotting armed shoppers. It's a PR thing, so people who are clueless will "feel good".
I remember having a somewhat heated discussion with a gal about concealed carry permits, and she kept saying things like "I don't want people with guns on my street", or, "why would I want people with guns near my kid's school?", as if a law against concealed carry would actually prevent it from happening. Gee, let's outlaw car accidents, while we're at it!