There have been several private-currancies in the recent past. One of them was written up in Utne Reader, quoting the Whole Earth Review.
These are the LETS systems, Local Exchange Transfer Systems. They seem to have been most successful in places of high unemployemnt as a way of increasing liquidity for services (mostly).
Inflation was not a problem because the money supply remained at zero.
The most telling remark from the originator (a Canadian) was that the system worked best when you had someone with deep pockets who was willing to run up a big positive balance by trading away things of value for the private currancy.
So it seems that the money supply, that is, the amount of liquidity available in the system, is not zero, but something else. There certainly are some interesting questions here, in particular the effective exchange rate between the national and local units of value. Eric