The most studied aspect of statistical network models is their degree structure, reflecting the propensity of nodes within a network to form connections with other nodes. Here we give a statistician's answer to the question of how to view network degree distributions as summary statistics, and thus how to model them. We do this by way of introducing a class of hierarchical statistical models for simple random graphs, to understand the scaling and heterogeneity properties of network degrees across varying realizations and sample sizes. Inspired by related work in probability and random graph theory, our construction is based on random weights that give rise to degree sequences through independent Bernoulli trials. We quantify the variation of degree sequences within and across networks in this setting, providing exact distributional results, limit theorems, and large-sample approximations governing the behavior of models in this class. Our results provide new tools to understand the heterogeneity of network degrees observed in practice, and to explore how properties of degree sequences, power law and otherwise, scale with network size.