Learning from data owned by several parties, as in federated learning, raises challenges regarding the privacy guarantees provided to participants and the correctness of the computation in the presence of malicious parties. We tackle these challenges in the context of distributed averaging, an essential building block of distributed and federated learning. Our first contribution is a novel distributed differentially private protocol which can match the accuracy of the trusted curator model even when each party communicates with only a logarithmic number of other parties chosen at random. Our second contribution is to enable users to prove the correctness of their computations without compromising the efficiency and privacy guarantees of the protocol. Our construction relies on standard cryptographic primitives like commitment schemes and zero knowledge proofs.