  We propose a new \cu{class-optimal} algorithm for the distributed computation
of Wasserstein Barycenters over networks. Assuming that each node in a graph
has a probability distribution, we prove that every node can reach the
barycenter of all distributions held in the network by using local interactions
compliant with the topology of the graph. We provide an estimate for the
minimum number of communication rounds required for the proposed method to
achieve arbitrary relative precision both in the optimality of the solution and
the consensus among all agents for undirected fixed networks.
