MUTE File Sharing is a peer-to-peer network that provides easy search-and-download functionality while also protecting your privacy.

Freenet is free software which lets you publish and obtain information on the Internet without fear of censorship. To achieve this freedom, the network is entirely decentralized and publishers and consumers of information are anonymous. Without anonymity

GNUnet is a framework for secure peer-to-peer networking that does not use any centralized or otherwise trusted services.

For the complete novice, sharing on BitTorrent doesn’t come any easier than this.