In Florida...
...there does not seem to be any particular firearm-related law prohibiting carrying on private property. It looks, from what I can see, like reliance is made on trespassing law. In other words, if the property owner or his authorized agent determines you are carrying and asks you to leave, you must. But you haven't committed a crime until you refuse to leave.
I would guess that would mean that a business could post a sign on the door saying they don't allow firearms and that plus your presence with a concealed weapon could be construed as tresspassing. So, just in case that's the case, I am starting to develop the habit of looking for such signs when entering businesses.
However, some businesses have so many signs that you could probably argue that you didn't notice the sign, so long as you left after being asked.
I have never noticed a sign. But if I do, I won't go inside unless I absolutely have to, in which case I'll go unarmed. Then, if possible, I'll try to find another source of what I went in for. But as for the bank and the post office, well, luckily I don't go to either place a lot.