The only thing the peer to peer network would do is escrow the transaction by not allowing one side of the deal to go through without the other also going through.
To give you an example: Let's say you agree to buy ฿10 from me for $150 USD, and you're going to pay me via PayPal.
I like the approach you're describing, except my understanding of Paypal is that the risk of chargebacks would make this too risky for the seller (bitcoinmarket.com cut off PPUSD due to chargeback fraud, from what I hear -- a friend of mine was burned).
Also, to prevent the market from being flooded with unbacked buy orders, we would need a trusted escrow that could guarantee that sufficient USD to carry out the order is being held in reserve for each outstanding buy order in the network (otherwise we wouldn't have an accurate metric of market depth).