You would just need the issuers of the local paper currency to be able to show and prove backing for the paper currency in a publicly auditable fashion. This means the issuer(s) would have to put up the bitcoin first before issuing the local paper currency. The purchasing power of the local currency would fluctuate with the value of bitcoin unless you adjusted the amount of bitcoin backing regularly so that the local paper currency is always traded 1 to 1 with the US dollar.
This is the chief problem with this idea - until it becomes feasible to price shit directly in BTC, you're going to hard pressed to convince the non-internet-savvy to use it, which appears to be the chief goal of such an idea.
IMHO local currencies don't need Bitcoin - in exactly the same way that Bitcoin doesn't need any backing, local currencies do not either.
Bitcoin also does not need local currencies right now, or more specifically it doesn't really stand to benefit from them. A local paper currency backed with Bitcoin essentially undoes everything that's attractive about Bitcoin, which coupled with the price instability of BTC just makes the whole idea unworkable at the present time.
Bitcoin makes a fine bridge for things like ripple networks, because the actual worth of each currency can be compared, decided, and agreed upon at the time of the transaction. Giving out a paper note that has a wildly fluctuating value will turn the very people you're trying to attract off the idea, possibly permanently.