How would privately minted coins enter circulation?
I am unsure about the American situation, but I learned a bit of history about token issuance in pre-Confederation Canada. These comments are based on the notes in a token collector guide.
Since they were privately issued, they would be called “tokens” and not “coins,” as that was reserved for governmental issuance.
In the Canadian colonies, there was a deficiency of low denomination copper coinage, which were officially minted in Britain (and British governors did not really  care). So the locals took matters into their own hands.
The banks were serious about coinage, and the metal content was close to what was implied by face value. Non-banks also issued tokens, but the metal content was quite often below face value. The banks did not like these tokens, and they typically did not accept them at face value, and melted them down. (This means that the tokens with the least copper are the rarest, and worth the most now.)
The tokens entered circulation via being handed out by the firms that minted them. Even if the tokens were too dodgy to be used elsewhere, people could return them to the same shop.