This paper focuses on distributed state estimation for sensor network observing a discrete-time linear system. The provided solution is based on a Distributed Moving Horizon Estimation (DMHE) algorithm with consensus-based arrival cost and a pre-estimating Luenberger observer in the formulation of the local problem solved by each sensor. This leads to reduce the computation load, while preserving the accuracy of the estimation. Moreover, observability properties of local sensors are used for tuning the weights related to consensus information fusion built on an observability rank-based condition, in order to improve the convergence of the estimation error. Results obtained by Monte Carlo simulations are provided to compare the performance with existing approaches, in terms of accuracy of the estimations and computation time.