We investigate the problem of distributed state estimation of a linear time-invariant (LTI) system by a network of sensors. We propose a new approach to designing distributed observers based on the following intuition: a given node (sensor) can reconstruct a certain portion of the state solely by using its own measurements together with an appropriate Luenberger observer. Hence it only needs to rely on information obtained from neighbors for estimating the portion of the state that is not locally detectable. We build on this intuition in this paper by extending the idea of the Kalman observable canonical decomposition to a setting with multiple sensors. We then construct local Luenberger observers at each node based on this decomposition, and use consensus dynamics to estimate the unobservable portions of the state at each node. This leads to an estimation scheme that achieves asymptotic state reconstruction at each node of the network for the most general class of LTI systems, sensor network topologies and sensor measurement structures.