Peer-to-peer (P2P) systems are predominantly used to distribute trust, increase availability and improve performance. A number of content-sharing P2P systems, for file-sharing applications (e.g., BitTorrent and Storj) and more recent peer-assisted CDNs (e.g., Akamai Netsession), are finding wide deployment. A major security concern with content-sharing P2P systems is the risk of long-term traffic analysis — a widely accepted challenge with few known solutions. In this paper, we propose a new approach to protecting against persistent, global traffic analysis in P2P contentsharing systems. Our approach advocates for hiding data access patterns, making P2P systems oblivious. We propose OBLIVP2P— a construction for a scalable distributed ORAM protocol, usable in a real P2P setting. Our protocol achieves the following results. First, we show that our construction retains the (linear) scalability of the original P2P network w.r.t the number of peers. Second, our experiments simulating about 16,384 peers on 15 Deterlab nodes can process up to 7 requests of 512KB each per second, suggesting usability in moderately latency-sensitive applications as-is. The bottlenecks remaining are purely computational (not bandwidth). Third, our experiments confirm that in our construction, no centralized infrastructure is a bottleneck — essentially, ensuring that the network and computational overheads can be completely offloaded to the P2P network. Finally, our construction is highly parallelizable, which implies that remaining computational bottlenecks can be drastically reduced if OBLIVP2P is deployed on a network with many real machines.