Social influence cannot be identified from purely observational data on social networks, because such influence is generically confounded with latent homophily, i.e., with a node's network partners being informative about the node's attributes and therefore its behavior. If the network grows according to either a latent community (stochastic block) model, or a continuous latent space model, then latent homophilous attributes can be consistently estimated from the global pattern of social ties. We show that, for common versions of those two network models, these estimates are so informative that controlling for estimated attributes allows for asymptotically unbiased and consistent estimation of social-influence effects in linear models. In particular, the bias shrinks at a rate which directly reflects how much information the network provides about the latent attributes. These are the first results on the consistent non-experimental estimation of social-influence effects in the presence of latent homophily, and we discuss the prospects for generalizing them.