A private business that does not allow weapons can turn it into a trespassing issue if one was not to leave after being asked. They are required to give you a warning of trespassing if one was to refuse at that point the law can be called and one can be arrested for trespassing. As far as a legal sign private business are not required to post any sign if they do not allow firearms. State and federal facilities are, if they do not allow firearms.
This is my simplified mode of action any private business sign or no sign I walk in. If they ask me to leave which has be only been twice in the past 3 years I leave. By law, private business must ask you to leave and then warn of trespassing before the law can get involved. For me to make it easier I just do not carry into most posted government facilities.