We investigate the performance of distributed estimation of static parameters in networks whose nodes make correlated measurements. The measurement models are linear with node specific time-varying observation matrices. The nodes cooperate with their neighbors by exchanging information that allow for approximation of the sufficient statistics in the estimation. We prove that the proposed estimation method is efficient. Specifically, we show that the performances of the distributed estimator and the centralized one are asymptotically the same, i.e., the limit of the ratio of their variances is 1.