We study the interaction between network effects and external incentives on file sharing behavior in Peer-to-Peer (P2P) networks. Many current or envisioned P2P networks reward individuals for sharing files, via financial incentives or social recognition. Peers weigh this reward against the cost of sharing incurred when others download the shared file. As a result, if other nearby nodes share files as well, the cost to an individual node decreases. Such positive network sharing effects can be expected to increase the rate of peers who share files. In this paper, we formulate a natural model for the network effects of sharing behavior, which we term the "demand model." We prove that the model has desirable diminishing returns properties, meaning that the network benefit of increasing payments decreases when the payments are already high. This result holds quite generally, for submodular objective functions on the part of the network operator. In fact, we show a stronger result: the demand model leads to a "coverage process," meaning that there is a distribution over graphs such that reachability under this distribution exactly captures the joint distribution of nodes which end up sharing. The existence of such distributions has advantages in simulating and estimating the performance of the system. We establish this result via a general theorem characterizing which types of models lead to coverage processes, and also show that all coverage processes possess the desirable submodular properties. We complement our theoretical results with experiments on several real-world P2P topologies. We compare our model quantitatively against more naïve models ignoring network effects. A main outcome of the experiments is that a good incentive scheme should make the reward dependent on a node's degree in the network.