We consider the task of recovering correlated vectors at a central decoder based on fixed linear measurements obtained by distributed sensors. A general formulation of the problem is proposed, under both a universal and an almost sure reconstruction requirement. We then study a specific correlation model which involves a filter that is sparse in the time domain. While this sparsity assumption does not allow reducing the description cost in the universal case, we show that large gains can be achieved in the almost sure scenario by means of a novel distributed scheme based on annihilating filters. The robustness of the proposed method is also investigated.