Consider that an autonomous linear time-invariant (LTI) plant is given and that each member of a network of LTI observers accesses a portion of the output of the plant. The dissemination of information within the network is dictated by a pre-specified directed graph in which each vertex represents an observer. This paper proposes a distributed estimation scheme that is a natural generalization of consensus in which each observer computes its own state estimate of the plant using only the portion of the output vector accessible to it and the state estimates of other observers that, according to the graph, are available to it. Unlike straightforward high-order solutions in which each observer broadcasts its measurements throughout the network, the average dimension of the state of each observer in the proposed scheme does not exceed the order of the plant plus one. We determine necessary and sufficient conditions for the existence of a parameter choice for which the proposed scheme attains asymptotic omniscience of the state of the plant at all observers. The conditions reduce to certain detectability requirements that imply that if omniscience is not possible under the proposed scheme then it is not viable under any other scheme-including higher-order LTI, nonlinear, and time-varying ones-subject to the same graph.