This sort of privatization of money is precisely what F.A. Hayek argues for in his masterful Denationalisation of Money.
Hayek wrote this near the end of his career, after thinking through all the economic arguments for monetary reform and examining the political viability of various proposals. He shows the essential nonviability of government money, and calls for a completely free market in the production, distribution, and management of money.
This book is the very core of the Hayekian approach to monetary policy, and the book that drew the world's attention to this radical thinker following his Nobel Prize in economics. The argument is substantively similar to Mises's, but rather than arguing for a gold standard, Hayek argues for utter abandonment of governmental attempts to reform money. The result: competitive private currencies that permit the market alone to choose the dominant currency the world over.
In the digital age, his argument takes on new significance, as experimentation in digital currencies continues apace and the harm of inflation takes its toll in new and painful ways.
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