In this paper, we present a model of distributed parameter estimation in networks, where agents have access to partially informative measurements over time. Each agent faces a local identification problem, in the sense that it cannot consistently estimate the parameter in isolation. We prove that, despite local identification problems, if agents update their estimates recursively as a function of their neighbors' beliefs, they can consistently estimate the true parameter provided that the communication network is strongly connected; that is, there exists an information path between any two agents in the network. We also show that the estimates of all agents are asymptotically normally distributed. Finally, we compute5 the asymptotic variance of the agents' estimates in terms of their observation models and the network topology, and provide conditions under which the distributed estimators are as efficient as any centralized estimator.