IP Addresses and Internet Privacy
I guess the real question is whether this is actually a problem or not. I would say that the average user has very little to worry about at the moment. As you may have noted, IP addresses have really only been linked with their real owners in legal matters. This is basically just the risk of piracy, so either stop pirating, accept the risk or find a suitable workaround.
The average person can't really do much with an IP address except run it through an IP lookup program. An IP lookup will only give the person's general area, operating system and web browser type. As I said, administrators will probably only bother to go through the logs if you manage to get yourself banned from the site, or they'll use the information to have a rough idea of where their traffic comes from and what browsers they use so that the website can be better optimized.
If you're very concerned about the issue, then you can look into using Tor, which is designed to hide your IP address. We have a full article on Tor's encryption methods if you're interested.
My final note would be that you should probably not even consider using a web proxy, unless you're the owner. Web proxies promise Internet browsing within a website, which effectively hides your IP address and makes it harder for anyone monitoring your connection to see what you're really doing. If you're actually concerned about your privacy though, note that the owner can see everywhere you went, when you went there and where you really are. This is coming from an ex-proxy owner too. Despite what they promise, they are far from the answer to Internet privacy.