A lot of times they just sell it for whatever they have in it for time and parts. I've bought a few engine blocks, heads etc this way from machine shops. People drop them off for machine work and then never come get em. I end paying less by a couple hundred dollars and can find good sets of heads and blocks from way back when that otherwise are pricey and the machine work is already done.
[quote/]Say someone leaves a car in a public place, or some third party's private property for a while, no for sale signs no nothing.
Assuming you mean a public roadway or something to that effect.
I don't know if a citizen could claim it or not. Here in Beaumont we just mark em for storage and after 96 hrs they are towed by a non preference wrecker. If they're a traffic problem then they get towed right then and it's between the vehicle owner and wrecker company from then on.