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We introduce a distributed adaptive algorithm for linear minimum mean squared error (MMSE) estimation of node-specific signals in a fully connected broadcasting sensor network where the nodes collect multichannel sensor signal observations. We assume that the node-specific signals to be estimated share a common latent signal subspace with a dimension that is small compared to the number of available sensor channels at each node. In this case, the algorithm can significantly reduce the required communication bandwidth and still provide the same optimal linear MMSE estimators as the centralized case. Furthermore, the computational load at each node is smaller than in a centralized architecture in which all computations are performed in a single fusion center. We consider the case where nodes update their parameters in a sequential round robin fashion. Numerical simulations support the theoretical results. Because of its adaptive nature, the algorithm is suited for real-time signal estimation in dynamic environments, such as speech enhancement with acoustic sensor networks.