We propose a new model, the Neighbor Mixture Model (NMM), for modeling node labels in a graph. This model aims to capture correlations between the labels of nodes in a local neighborhood. We carefully design the model so it could be an alternative to a Markov Random Field but with more affordable computations. In particular, drawing samples and evaluating marginal probabilities of single labels can be done in linear time. To scale computations to large graphs, we devise a variational approximation without introducing extra parameters. We further use graph neural networks (GNNs) to parameterize the NMM, which reduces the number of learnable parameters while allowing expressive representation learning. The proposed model can be either fit directly to large observed graphs or used to enable scalable inference that preserves correlations for other distributions such as deep generative graph models. Across a diverse set of node classification, image denoising, and link prediction tasks, we show our proposed NMM advances the state-of-the-art in modeling real-world labeled graphs.