It has been recently shown in Ren et al. (2010) that by collecting noise-contaminated time series generated by a coupled-oscillator system at each node of a network, it is possible to robustly reconstruct its topology, i.e. determine the graph Laplacian. Restricting ourselves to linear consensus dynamics over undirected communication networks, in this paper we introduce a new dynamic average consensus least-squares algorithm to locally estimate these time series at each node, thus making the reconstruction process fully distributed and more easily applicable in the real world. We also propose a novel efficient method for separating the off-diagonal entries of the reconstructed Laplacian, and examine several concepts related to the trace of the dynamic correlation matrix of the coupled single integrators, which is a distinctive element of our network reconstruction method. The theory is illustrated with examples from computer, power and transportation systems.