the asymptotic variance of the estimator proposed in this paper reveals the scaling with respect to the number of agents interested in reconstructing a particular entry of the parameter. A field estimation scheme in a fully distributed setup of the type studied in this paper with arbitrary connected inter-agent communication topology where agents reconstruct only a subset of the physical field was also proposed in [@kar2010large] (Chapter 3). The current work is inspired by [@kar2010large] and generalizes the development in [@kar2010large] in several fronts to achieve better estimation performance. In [@kar2010large], a *single time-scale* consensus+innovations algorithm pertaining to a linear observation model was proposed and the consistency and asymptotic normality[^2] of the estimator was established. By, a *single time-scale* consensus+innovations algorithm, we mean algorithms where the consensus and innovation potentials are controlled by the same time-decaying sequence. The performance of the single time-scale version of the consensus+innovations distributed estimation algorithm in terms of asymptotic variance depends on the network topology and is thus affected when the connectivity of the network is relatively poor. In contrast with [@kar2010large], we propose a $\emph{consensus}+\emph{innovations}$ algorithm, where the *consensus* and *innovations* terms are weighed through different carefully crafted time-varying sequences. In this paper, we 