Take a look at OpenTransactions (
https://github.com/FellowTraveler/Open-Transactions). It's a platform that FellowTraveller is developing to enable easily setting up private banking, which can be conveniently backed and cleared by BTC. If someone wanted to then issue paper money that was redeemable for their private bank, that would be up to them (it's risky, since paper money is really hard to make unforgeable; you basically have to be big enough to absorb the losses when people eventually do forge it.) But FT's solution gives you instant clearing, as well as double blinding, so it's truly anonymous (as compared to Bitcoin, which is only as anonymous as you are able to obscure the connection between your addresses and your identity.)