This paper deals with distributed state estimation for a class of networked systems, which extends the results of Zhou (Coordinated one-step optimal distributed state prediction for a networked dynamical system, IEEE Transactions on Automatic Control, 58(11): 2756-2771, 2013) on one-step state predictions. Through introducing a specific discrete state-space representation of the distributed estimator, whose number of internal inputs and outputs are twice as that of the plant, recursive and explicit expressions are derived respectively for the optimal gain matrix and the covariance matrix. Their expressions have similar structures to those of the state predictor obtained in Zhou (2013), which means that the desired state estimator inherits the computational advantages of the state predictor, and is therefore more suitable for systems consisting of a large number of subsystems than the lumped Kalman filter. Numerical simulations show that the state estimator developed in this paper may even have the same estimation accuracy as that of the lumped Kalman filter.