In this paper, we study a distributed parameter estimation problem in a large-scale network of communication sensors. The goal of the sensors is to find a global estimate of an unknown parameter minimizing, which minimizes some aggregate cost function. Each sensor can communicated to a few “neighbors”, furthermore, the communication channels have limited capacities. To solve the resulting optimization problem, we use a weighted modification of the distributed consensus-based SPSA algorithm whose main advantage over the alternative method is its ability to work in presence of arbitrary unknown-but-bounded noises whose statistical characteristics can be unknown. We provide a convergence analysis of the weighted SPSA-based consensus algorithm and show its efficiency via numerical simulations.