To my knowledge no distro uses peer-to-peer for downloading packages. If you have lots of computers running the same Linux distribution most package management systems allow you to run your own repositories and mirrors. This is useful if you have a slow internet connection or if you have a very large amount of machines running the same distribution.
I did a quick google search for this, however, and I found apt-p2p. This looks like a peer-to-peer solution for Debian (and possibly Debian derivatives like Ubuntu) repositories. I have never tried this, but it might be worth looking into.
Edit: I also found p2pacman for Arch Linux, this piece of software seems to be more of an experiment than something ready for wide usage.