How do private banks coin money?
There would be no laws stopping them from doing so. Of course, history shows people prefer paper currency fiduciary notes over coins, since the latter are bulky as well as hard to store and transport. Money would be coined, if at all, by private mints and this would probably function no differently than coinage does today, though it would be more demand specific. I.e., coins would not be printed willy-nilly, but mostly based on consumer demand for them. Likewise, coins would have to be evaluated on their fineness and the like. Chances are, under free banking, coined money would not play a major role in the money system.