For the record however I belive they do not. That is arguable based on if they are in fact private property. But even if they are, I would challenge the premise that private property that is open to the public gives the owner the discretion of wether or not CODIFIED rights can be restricted.
While the owner of a private property cannot take legal action against you for violating policies that are in conflict with COTUS, he or she may expel you from that property for any reason that he or she sees fit. The proprieter of most businesses is well within his rights to expel you from his business if you exercise your 1st Amendment rights in a vulgar or offensive fashion that he takes issue with. I see no reason why the same would not apply to the 2nd amendment. Like I said, I am totally against a college having you arrested for carrying a gun on campus, but I think that it is within their rights, whether we agree with it or not, to expel you from their campus if you choose to violate their no-firearms policy.