There is a new model and NBER paper to come from Jesus Fernandez-Villaverde and Daniel Sanches (pdf, ungated), here is the abstract:
Can competition among privately issued fiat currencies such as Bitcoin or Ethereum work? Only sometimes. To show this, we build a model of competition among privately issued at currencies. We modify the current workhorse of monetary economics, the Lagos-Wright environment, by including entrepreneurs who can issue their own fiat currencies in order to maximize their utility. Otherwise, the model is standard. We show that there exists an equilibrium in which price stability is consistent with competing private monies, but also that there exists a continuum of equilibrium trajectories with the property that the value of private currencies monotonically converges to zero. These latter equilibria disappear, however, when we introduce productive capital. We also investigate the properties of hybrid monetary arrangements with private and government monies, of automata issuing money, and the role of network effects.
I would stress a few points. First, the world is going to have some form of currency competition whether one likes it or not. That is already the case today, so these are very real questions, not just thought games for libertarians.
Second, the Bitcoin and broader cryptocurrency experience indicate that the marginal cost of issuing a new private currency is well above zero, contra some of the literature from the 1970s, which viewed the enterprise in terms of printing money and paying only for the additional paper. Bitcoin can be interpreted as a commodity currency of sorts, where the relevant expenditures are on codebreaking and electricity, rather than digging up gold from the ground. Once it is focal enough, it might be able to provide some version of rough price stability in terms of its unit.
Third, if your government is halfway legitimate and not broke, its currency is likely to be a dominant winner in these forms of currency competition, especially to the extent that currency is supported by the fiscal authority. In this sense it is almost impossible to get away from a legitimate or even semi-legitimate government-issued currency.