Sorry, but no. Coinage was the right of kings for millenia. According to historians, the entire history of using coins as money started with governments basically inventing the use of coins as a technique for simplifying the logistics of their armies (soldiers would be given coins that they could use to pay local populations for food; in exchange, the government would accept the coins instead of physical goods as payment of taxes).

Even during the period of largely unregulated private banking that reached its peak in the 19th century, the private banks largely just issued paper money that was denominated in coins that were issued by governments.