In this paper, the design methods and fundamental performance analysis of an adaptive peer-to-peer estimator are established for networks exhibiting message losses. Based on a signal state model, estimates are locally computed at each node of the network by adaptively filtering neighboring nodes' estimates and measurements communicated over lossy channels. The computation is based on a distributed optimization approach that guarantees the stability of the estimator while minimizing the estimation error variance. Fundamental performance limitations of the estimator are established based on the variance of the estimation error in relation to the message loss process. Numerical simulations validate the theoretical analysis and illustrate the performance with respect to other estimators. (C) 2015 Elsevier Ltd. All rights reserved.