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This work presents a distributed algorithm for observer design for linear continuous-time systems. We assume the dynamical system to be partitioned into disjoint areas, and we let each area be equipped with a control center. Each control center knows local dynamics, collects local observations, performs local computation, and communicates with neighboring control centers at discrete times. For our continuous-discrete estimation algorithm we prove convergence, we characterize its convergence rate, and we show robustness against discretization and communication errors. Our technical approach is inspired by waveform relaxation methods and combines tools from estimation theory, decentralized control theory, and parallel computation. We illustrate the effectiveness of our algorithm with illustrative examples in sensor networks and electric power systems.