I know that the local police refuse to investigate an accident in a parking lot. I have had several hit-and-run accidents which did significant damage to my vehicle. When I called the police, they told me, in as many words, that they do not investigate accidents “on private property.”
I would assume, unless they were stationed in the parking area with the express purpose of catching violators, that they also would not ticket Stop-sign runners.
As stated it varies widely from state to state. I am a police officer in NJ. Most of the traffic laws (Title 39 in NJ) are not enforceable on private property. There are off the top of my head 7 statutes that are. Stop signs on private property are not. However careless driving is so running a post stop sign in a parking lot could easily be proved to be careless. And in some parking lots the owner has asked for enforcement in their lot. In those cases the local government can authorize that Title 39 be enforced on that particular property. That happens often on private roads such as in apartment complexes. Each state will have specific circumstances such as that.
In the 1980s there was a case in Williamsport PA where a man ran a stop sign in a store parking lot and hit someone else.
In court, his defense was that he was on private land and therefore the stop sign could not be legally enforced. The judge ruled that it was a public thoroughfare and therefore he could be expected to obey the laws of the road.
The case was appealed, but I do not know how the appeal turned out.