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We consider the problem of distributing k blocks from a source to N nodes in a peer-to-peer (P2P) network with both node upload and download capacity constraints. As k scales up, we prove for homogeneous networks that if network coding is allowed, randomly matching senders and receivers in each time slot asymptotically achieves the maximum downloading rate at each node. For heterogeneous networks with network coding allowed, we show that a fair and optimal downloading rate at each node can be asymptotically approached, if in each time slot, each node randomly allocates its upload bandwidth to its receivers that have available download bandwidth. We also give a performance lower bound of the above randomized coded dissemination when both k and N scale under certain conditions. These results demonstrate that with network coding, simple randomized receiver selection and rate allocation suffice to achieve P2P broadcast capacity, forming a theoretical foundation for mesh-based P2P networks with network coding.