In this paper we consider a nonlinear constrained system observed by a sensor network and propose a distributed state estimation scheme based on Moving Horizon Estimation (MHE). In order to embrace the case where the whole system state cannot be reconstructed from data available to individual sensors, we resort to the notion of MHE-detectability for nonlinear systems, and add to the MHE problems solved by each sensor a consensus term for propagating information about estimates through the network. Under some suitable assumptions we prove convergence to zero and stability of the state estimation error provided by any sensor.