In the Wall Street Journal, Steve Hanke discusses the idea of eliminating the $1 bill, which has been proposed on the grounds that $1 coins are cheaper and would save the federal government money. He proposes instead that private issuers be allowed to issue $1 bills. People can then decide whether they prefer government $1 coins to private $1 bills. As I showed in a Cato Journal article some years ago, a provision in a reform of U.S. banking law in 1994 made it is legal for U.S. banks to issue notes. It is unclear whether the drafters of the law were aware that they were (re)legalizing private note issue. So far no banks have issued notes. At the Cato Institute's annual monetary conference a couple of years ago when the subject was discussed, my impression was that would-be private issuers of notes want additional assurance. The law lets them issue notes but they are concerned about the possibility of regulatory or other forms of harassment for doing so.
George Selgin has advocated privatizing the production of pennies, which the federal government has also talked about eliminating. Incidentally, from a quick review of the U.S. Code I did not see any explicit provision that minting coins is a government monopoly, though maybe I missed it and a reader can enlighten me. The monopoly has been enforced through the counterfeiting laws, including in the Liberty Dollar case.