I consider public building just as you have said, a government-run building which is available to the public as a service or housing folks who provide public service. The Post Office, for example.
But, I also don't know of any privately owned establishments that have a weapon screening at the entrance. If there were, I wouldn't want to patronize their business anyway, much less go in there.
Private establishments, as you said, do have the right to decline service and entry to you based upon your carrying a weapon, however. I have seen places with signs in our state here barring patrons who carry a concealed weapon. I don't generally patronize them.
But, if they don't have sufficient signage of their policy, all they can do is ask you to leave, if for some reason they become aware of your concealed weapon.
If someone gets arrested at a mall for legally carring a concealed weapon, just for having the weapon, without having been asked to leave first, I would like to see them take action not only against the property, but the security or law enforcement agency who processed the arrest as well. Here in Colorado, an establishment who does not wish to have licensed people on their premisis must 1. give proper and ample signage notifying folks who enter of their policy, and 2. upon becoming aware of the concealed carrier, must ask them to leave first...all before calling the police and/or having them arrested or charged with tresspassing.