To get FIAT into a P2P exchange you must first get a P2P representation of fiat value. We know that crypto currencies convey value. Therefore a crypto-fiat currency that is issued as fiat-(IO-the-network) backed by crypto-collateral (held by the network) ensurs that the crypto-fiat can be redeemed because it has more crypto-currency backing than fiat value. The key is that it doesn't matter who redeems it. All that matters is that whoever issues it locks enough crypto-currency in 'escrow' and that 'escrow payments' are automatically distributed to all holders of the fiat-crypto balances with each new block.
With this in mind you can find anyone in your area who wants to exchange crypto-USD for real USD and you can meet up and make the exchange. Alternatively you could use systems like dwolla or intra-bank transfers or money orders. Combine this with something like NashX mutually assured destruction, escrow, multi-sig, etc and you have effective conversion between fiat-crypto and fiat at near face value and no 'exchange spread' like you get with localbitcoins. Both parties are essentially exchanging equal value for equal value.
I posted a white-paper on another thread you may want to check out:
https://bitcointalksearch.org/topic/m.2238954A quick summary is this:
1) all balances (fiat or otherwise) pay interest
2) all fiat balances are created by someone borrowing against a crypto-currency used as collateral.
3) payments from the collateral are distributed to those who hold a balance in that fiat currency.
4) people borrow against their crypto-currency because they expect it to go up in value relative to fiat AND the crypto-currency also pays dividends from transaction fees.
5) once you have a fiat balance it can be traded as easily as a bitcoin balance.
6) fiat balances are always 100% redeemable at face value (excluding transaction fees) so long as the underlying crypto-currency maintains any amount of value.
7) the effective interest rate on fiat balances goes UP when the value of the crypto currency goes up.
Given my approach we can now eliminate the centralized exchange. Allow individuals acting as 'local-bitcoin operators' or posting on craigslist to 'deposit' and 'withdraw' $USD or EUR from other individuals without having exchange risk but only a potential 'transaction fee'. There would be more people wanting to deposit $USD because their $USD balances pay interest and with more people wanting to deposit that means more people wanting to withdraw which equals more opportunities for local exchanges. People could use the system without ever owning an asset denominated in a crypto-currency to trade $USD among themselves, etc.
The problem with fiat isn't the fiat, but the laws. The way you fix laws is to make breaking them so easy and hard to find and desirable to so many people that they effectively nullify the law. Copyright is effectively nullified by mass law-breaking and that is enabled by the internet. Sure some people get caught, but most do not. There are so many people doing it because it is 'profitable' to them that attempts to shut it down will be fruitless.