Agents in a social network learn about the true state of the world over time from their own signals and reports from immediate neighbors. Each agent only knows her local network, consisting of her neighbors and any connections among them. In each period, every agent updates her own estimates about the state distribution based on her perceived new information. She also forms estimates about each neighbor's estimates given the new information she thinks the neighbor has received. Whenever a neighbor's report differs from the agent's estimates of his estimates, the agent attributes the difference to new information. The agents form the correct Bayesian posterior beliefs in any network if  their information structures are partitional. They can also do so for more general information structures if the network is a social quilt, a tree-like union of completely connected subgroups. Under this procedure, the agents make fewer mistakes than under myopic learning; and they learn correctly if the network is common knowledge.