The leading approach for image compression with artificial neural networks (ANNs) is to learn a nonlinear transform and a fixed entropy model that are optimized for rate-distortion performance. We show that this approach can be significantly improved by incorporating spatially local, image-dependent entropy models. The key insight is that existing ANN-based methods learn an entropy model that is shared between the encoder and decoder, but they do not transmit any side information that would allow the model to adapt to the structure of a specific image. We present a method for augmenting ANN-based image coders with image-dependent side information that leads to a 17.8% rate reduction over a state-of-the-art ANN-based baseline model on a standard evaluation set, and 70-98% reductions on images with low visual complexity that are poorly captured by a fixed, global entropy model.