This paper focuses on distributed state estimation using a sensor network for monitoring a linear system. In order to account for physical constraints on process states and inputs, we propose a moving horizon approach where each sensor has to solve a quadratic programming problem at each time instant. We discuss conditions guaranteeing convergence of all estimates to a common value by characterizing the dynamics of the unobservable component of the state. Furthermore, we highlight how the performance of the state estimation scheme depends upon various observability properties of the system and discuss how different communication protocols impact on the quality of the estimates.