It seems that if you download a file via torrent, each downloader can see other downloaders IP addresses. I also noticed that my (windows) pc's name was sometimes displayed briefly.
How does one keep the IP private ?
DNS resolves domain names to IP addresses. IP addresses are the endpoint identifier so that TCP/IP can create a connection from the source computer to the destination computer, therefore has to be available for the protocol to use.
Hiding your IP in the final connection is like asking your mailman to use ESP to read where the letter is supposed to be delivered.
Things like TOR or properly set up proxy servers can be used to prevent the server end of the connection and most of the head of the chain from knowing the IP on the client end of the connection by passing messages between multiple computers and reassigning the origin addressing, but the TOR endpoint or proxy server still has to know your IP address in order to deliver the message on the final leg.
Using something like TOR with Torrents is next to impossible because of the many to many connections necessary in the seeding and transfer process. Multiple computers have to know your IP address in order to deliver the payload. No address, no delivery.
In real life, you hide your physical mail address by arranging for delivery to a CMRA, having someone pick it up from general delivery at the post office and putting it in a ground spike in your front lawn, renting a box at your local UPS store for package delivery, etc.
So, how would you do that with computers? Bit Torrent out of your local public access WiFi hotspot, use your Company computers to download and store the information, have someone else do the download and shove into a DropBox, etc. Some of these will have serious repercussions if they involve theft of services, but basically if you want to hide your IP, you do what TOR does, use someone else's IP.
Humor me by next asking how to hide your IP address from your ISP...
In addition to what Fiasco Labs said, there are a few VPN services names I could give you because I have used them myself that support P2P downloads, such as ivacy.com or iblocklist.com (phantompeer).
These VPN services do not have many options and often don't offer many VPN protocols, but at the very least you are guaranteed that they don't log any of your connection history or online activity. See this article and this article for more info.
You can't really keep your IP private if you're using a torrent on the open internet. The reason for this is the way the torrent protocol works. You aren't downloading directly from the server; instead, you are downloading from other downloaders. In order to get a file from said downloader, you must have some way to contact them - that is, an IP address or domain name.
There may be a possibility to use a privacy network such as TOR to avoid having your IP address found, but I don't know how to do it.