We consider model social networks in which information propagates directionally across layers of rational agents. Each agent makes a locally optimal estimate of the state of the world, and communicates this estimate to agents downstream. When agents receive information from the same source their estimates are correlated. We show that the resulting redundancy can lead to the loss of information about the state of the world across layers of the network, even when all agents have full knowledge of the network's structure. A simple algebraic condition identifies networks in which information loss occurs, and we show that all such networks must contain a particular network motif. We also study random networks asymptotically as the number of agents increases, and find a sharp transition in the probability of information loss at the point at which the number of agents in one layer exceeds the number in the previous layer.