Local network community detection aims to find a single community in a large network, while inspecting only a small part of that network around a given seed node. This is much cheaper than finding all communities in a network. Most methods for local community detection are formulated as ad hoc optimization problems. In this paper, we instead start from a generative model for networks with a community structure. By assuming that the network is uniform, we can approximate the structure of the unobserved parts of the network to obtain a method for local community detection. We apply this local approximation technique to two variants of the stochastic block model. This results in local community detection methods based on probabilistic models. Interestingly, in the limit, one of the proposed approximations corresponds to conductance, a popular metric in this field. Experiments on real and synthetic data sets show comparable or improved results compared to state-of-the-art local community detection algorithms.