Private Scottish (and Canadian) banks that issues bank notes in the 19th century did just fine catching counterfeiters. See https://www.cato.org/blog/more-counterfeit-currency

Law enforcement is a function that even many libertarians, but not all, reserve for the 'nanny'. See https://en.wikipedia.org/wiki/Night-watchman_state

Ahahahaha, 19th century banks had tiny circulation of their own Mickey Mouse notes, compare to global reach of the dollar. Now who would police the banks, lol who would arbitrary decline the notes in case of low liquidity?