Ah cool, I'm doing a degree on data communications, so that means thinking about this stuff is as good as studying
Well, for one thing, you can already do this to an extent. But you have to connect it to that other computer which already has a connection to the Internet. If you take it to an extreme, do note that it'd eat up a lot of CPU power and bandwidth. By asking the ISP to do it, you don't have to, and they handle all the traffic regulating. I, for one, don't want to forward someone's 3GB worth of porn packets.
One of the things an ISP traditionally does the physical part of the data transfer. Your telco handles the DSL stuff as well as other infrastructure. That's why some ISPs can't provide service for certain telcos. Now
someone will have to do this. You can regulate that someone, even if it's some public access line.
Security. All your packets going through random computers are going to be ridiculously easy to sniff. Encryption? Well, you'll still have to negotiate your ciphers and the negotiation will be going through multiple computers. It becomes much, much easier to sit around waiting for bank account data and cracking it in time.
Traffic. Today's Internet protocols should be able to handle it, IIRC, because they're already suspicious of everyone. Small rewrites are OK too. But yeah, you'll get bottlenecks. It's not like a physical highway. Some people will be grabbing ridiculous amounts of data and they'll jam up your laptop's network card.
Personally, I think this'd knock our capability back a generation, but it's not impossible. And should be cheaper if everyone does it. But right now, the evils of regulation don't exceed having a high-speed connection. After all, between non-censorship and porn that loads up faster, most people would pick faster porn (ironically).