In this work we present an approach for generating alternative text (or alt-text) descriptions for images shared on social media, specifically Twitter. More than just a special case of image captioning, alt-text is both more literally descriptive and context-specific. Also critically, images posted to Twitter are often accompanied by user-written text that despite not necessarily describing the image may provide useful context that if properly leveraged can be informative. We address this task with a multimodal model that conditions on both textual information from the associated social media post as well as visual signal from the image, and demonstrate that the utility of these two information sources stacks. We put forward a new dataset of 371k images paired with alt-text and tweets scraped from Twitter and evaluate on it across a variety of automated metrics as well as human evaluation. We show that our approach of conditioning on both tweet text and visual information significantly outperforms prior work, by more than 2x on BLEU@4.