We examine how well people learn when information is noisily relayed from person to person, subject to: dropping, mutation, and deliberate manipulation. This allows us to explore how communication platforms can improve learning without censoring or even examining messages, but purely by limiting the number of times a message can be relayed or the number of people to whom someone can forward a message. In particular, we analyze learning as a function of the network depth (length of relay chains) and breadth (how many chains a person has access to). Noise builds up as depth increases and so learning requires greater breadth, which we show to be characterized via a sharp threshold above which the receiver learns fully and below which the receiver learns nothing. Moreover, we show that small uncertainty about the rates of mis- and dis-information make learning from long chains of messages impossible. Optimizing learning requires either limiting depth (by controlling how many times a message can be forwarded), or if that is not possible then limiting breadth (by capping the number of people to whom someone can forward a message). Although limiting breadth decreases the overall amount of information a learner has access to, it increases the relative fraction of messages that are coming from nearby compared to far away in the network, and thus increases the signal to noise ratio. Such policies do not require the ability to fact-check, respect privacy, increase the fraction of true to false messages, and have been implemented by communication platforms. Finally, we extend our model to study learning from dropping rates (e.g., people are more likely to pass messages with one conclusion than another). We find that as the distance to primary sources grows, all learning comes from either the total number of messages received or from the content of received messages, but the learner does not need to pay attention to both.