Graph convolutional neural networks (GCNNs) have been shown to accurately predict materials properties by featurizing local atomic environments. However, such models have not yet been utilized for predicting per-site features such as Bader charge, magnetic moment, or site-projected band centers. In this work, we develop a per-site crystal graph convolutional neural network that predicts a wide array of per-site properties. This model outperforms a per-element average baseline, and is thus capturing the effect of the neighborhood around each atom. Using magnetic moments as a case study, we explore an example of underlying physics the per-site model is able to learn.