Trends and opinion mining in social media increasingly focus on novel interactions involving visual media, like images and short videos, in addition to text. In this work, we tackle the problem of visual sentiment analysis of social media images -- specifically, the prediction of image sentiment polarity. While previous work relied on manually labeled training sets, we propose an automated approach for building sentiment polarity classifiers based on a cross-modal distillation paradigm; starting from scraped multimodal (text + images) data, we train a student model on the visual modality based on the outputs of a textual teacher model that analyses the sentiment of the corresponding textual modality. We applied our method to randomly collected images crawled from Twitter over three months and produced, after automatic cleaning, a weakly-labeled dataset of $\sim$1.5 million images. Despite exploiting noisy labeled samples, our training pipeline produces classifiers showing strong generalization capabilities and outperforming the current state of the art on five manually labeled benchmarks for image sentiment polarity prediction.