Interesting idea. I think of your idea as a general p2p delivery system comprised of a network of pick-up points and drop-off points, whereby a person is compensated for delivering an item from a pick-up point to a drop-off point.
p2p restaurant deliveries seems like a bad idea because the deliveries are too time sensitive (food gets cold relatively quickly).
Why not partner with UPS/Fedex as a way to reduce some of their costs? So driving home from work, you stop by a local UPS/Fedex pick-up point where you are given a package to deliver to Fred, who lives a mile away from your home. (The local pick-up points can established such that all package have associated drop-off points within a X-mile radius.) Or for deliveries between airpots, people can be compensated (i.e. offered a discounted ticket) if they carry-on a package to be dropped off at their destination airport.
With the rising gas prices, I suspect many people would love to be compensated for going a few minutes out of their way during their regular commutes. Not to mention, the publicity this would generate is immense.