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Consensus in sensor networks is a procedure to corroborate the local measurements of the sensors with those of the surrounding nodes, and leads to a final agreement about a common value that, in detection applications, represents the decision statistic. As the amount of collected data increases, the convergence toward the final statistic is ruled by suitable scaling laws, and the question arises if the asymptotic (large sample) properties of a detection statistic are retained when this statistic is approximated via consensus algorithms. We investigate the asymptotic properties of running consensus detectors both under the Neyman-Pearson paradigm (fixed number of data) and in the sequential case. An appropriate asymptotic framework is developed, and exact theoretical results are provided, showing the asymptotic optimality of the running consensus detector. In addition, numerical experiments are performed to address nonasymptotic scenarios.