We study the problem of designing distributed functional observers for LTI systems. Specifically, we consider a setting consisting of a state vector that evolves over time according to a dynamical process. A set of nodes distributed over a communication network wish to collaboratively estimate certain functions of the state. We first show that classical existence conditions for the design of centralized functional observers do not directly translate to the distributed setting, due to the coupling that exists between the dynamics of the functions of interest and the diverse measurements at the various nodes. Accordingly, we design transformations that reveal such couplings and identify portions of the corresponding dynamics that are locally detectable at each sensor node. We provide sufficient conditions on the network, along with state estimate update and exchange rules for each node, that guarantee asymptotic reconstruction of the functions at each sensor node.