Optimization problems arising in signal and image processing involve an increasingly large number of variables. In addition to the curse of dimensionality, another difficulty to overcome is that the cost function usually reads as the sum of several loss/regularization terms, which are non-necessarily smooth and possibly composed with large-size linear operators. Proximal splitting approaches are fundamental tools to address such problems, with demonstrated efficiency in many applicative fields. In this paper, we present a new distributed algorithm for computing the proximity operator of a sum of non-necessarily smooth convex functions composed with arbitrary linear operators. Our algorithm relies on a primal-dual splitting strategy, and benefits from established convergence guaranties. Each involved function is associated with a node of a hypergraph, with the ability to communicate with neighboring nodes sharing the same hyperedge. Thanks to this structure, our method can be efficiently implemented on modern parallel computing architectures, distributing the computations on multiple nodes or machines, with controlled requirements for synchronization steps. Good numerical performance and scalability properties are demonstrated on a problem of video sequence denoising. Our code implemented in Julia is made available at \url{https://github.com/MarinENSTA/distributed_julia_denoising}.