Egocentric network sampling observes the network of interest from the point of view of a set of sampled actors, who provide information about themselves and anonymized information on their network neighbors. In survey research, this is often the most practical, and sometimes the only, way to observe certain classes of networks, with the sexual networks that underlie HIV transmission being the archetypal case. Although methods exist for recovering some descriptive network features, there is no rigorous and practical statistical foundation for estimation and inference for network models from such data. We identify a subclass of exponential-family random graph models (ERGMs) amenable to being estimated from egocentrically sampled network data, and apply pseudo-maximum-likelihood estimation to do so and to rigorously quantify the uncertainty of the estimates. For ERGMs parametrized to be invariant to network size, we describe a computationally tractable approach to this problem. We use this methodology to help understand persistent racial disparities in HIV prevalence in the US. We also discuss some extensions, including how our framework may be applied to triadic effects when data about ties among the respondent's neighbors are also collected.