In this paper, a novel protocol for providing both client and server anonymity in peer-to-peer (P2P) based grids is presented. The protocol assumes individual nodes or users to be utility maximizing agents, and relies on an auction mechanism for trading of resources among them. The resources here can refer to data files, storage capacity, or computation power (i.e., CPU cycles) etc. The protocol is inherently anonymous, light-weight, and incentive-compatible. Incentive compatibility implies that the protocol takes into account the selfishness of users; as would be seen the utilities of users are maximized by truthfully following the protocol steps. Moreover, if the protocol is truthfully followed by the users, anonymity to both the clients and servers of all the transactions are guaranteed. Furthermore, unlike other schemes, the proposed protocol does not rely on any trusted centralized entity or require specialized encryptions to be performed by the users. Thus, the protocol incurs very low overhead on the system and is light-weight. In brief, the proposed protocol provides efficient and natural means to provide anonymity in P2P based grids, and is easily deployable in a large un-trusted Internet-scale setting.