private property owners (like a mall . . .) can post "No Guns"
whether their posting carries any force of "law" depends on the state.
in some states "No Guns" is merely a suggestion. those states have no law that allows a private property owner to preemptively prohibit carry of a concealed weapon.
in most states if a private property owner tells you to leave (because of weapon, or other) and you refuse, the best they can muster is trespassing charges. without regard to whether they post "No Guns" or not . this is the Nevada casino defense. btw, in Nevada, do follow the casino's suggestions - the local courts understand with no doubt where the money is coming from, and when it comes to "you vs. casino" - you lose, not a single if about it.
NY - after having been severely beat about the head and face by SCOTUS, is busy outlining places where concealed weapons may not be carried . . . now, aside from bars, where drunk idiots go out of control regularly . . . NY has banned carrying a weapon anywhere you may need one. NY's restrictions will not stand - however it'll take several decades to work through the courts.