The paper addresses the media-specific rate allocation problem in multipath networks. The streaming rate on each path is determined such that the end-to-end media distortion is minimized, when the receiving client aggregates packets received via multiple network channels. As it is difficult for the media server to have the full knowledge about the network status, we propose a distributed path selection and rate allocation algorithm. The network nodes participate to the optimization strategy, based on their local view of the network status. This eliminates the need for end-to-end network monitoring, and allows for the deployment of large scale rate allocation solutions. We design an optimal rate allocation algorithm, where the media client iteratively updates the best set of streaming paths. According to this rate allocation, each intermediate nodes then forwards incoming media flows on the outgoing paths, in a distributed manner. The proposed algorithm is shown to quickly converge to the optimal rate allocation solution, and hence to lead to stable rate allocation solutions. We also propose a greedy distributed algorithm that achieves close-to-optimal end-to-end distortion performance in a single pass. Both algorithms are shown to outperform simple heuristic-based rate allocation approaches for numerous random network topologies, and therefore offer an interesting solution for media-specific rate allocation over large scale multi-path networks.