I have read that the actuality includes a loophole you didn't speak of:
Once someoen owns shares, they can privately-sell them, or give them away, or will them to someone..
Once enough people have done this, the "private" company becomes actually publically-traded, though not on any exchange..
..creating some legal difficulties, re regulations.
From that bit, which I never would have known to even consider ( some article I read, some years ago ), then it looks like people can sell their shares to another private-individual.
Maybe some jurisdictions prohibit that.
I don't know, I'm just identifying an angle people apparently haven't commonly considered.