The paper deals with the problem of estimating the state of a linear system over a peer-to-peer network of linear sensors. The proposed approach is fully distributed, scalable, and allows for taking into account constraints on noise and state variables by resorting to the moving-horizon estimation paradigm. Each network node computes its local state estimate by minimizing a cost function defined over a sliding window of fixed size. The cost function includes a fused arrival cost, which is computed in a distributed way by performing a consensus on the local arrival costs. The proposed estimator guarantees stability of the estimation error dynamics in all network nodes, under the minimal requirements of network connectivity and collective observability, and for any number of consensus steps. Numerical simulations are provided to demonstrate the practical effectiveness of the approach.