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How can we teach robots or virtual assistants to gesture naturally? Can we go further and adapt the gesturing style to follow a specific speaker? Gestures that are naturally timed with corresponding speech during human communication are called co-speech gestures. A key challenge, called gesture style transfer, is to learn a model that generates these gestures for a speaking agent 'A' in the gesturing style of a target speaker 'B'. A secondary goal is to simultaneously learn to generate co-speech gestures for multiple speakers while remembering what is unique about each speaker. We call this challenge style preservation. In this paper, we propose a new model, named Mix-StAGE, which trains a single model for multiple speakers while learning unique style embeddings for each speaker's gestures in an end-to-end manner. A novelty of Mix-StAGE is to learn a mixture of generative models which allows for conditioning on the unique gesture style of each speaker. As Mix-StAGE disentangles style and content of gestures, gesturing styles for the same input speech can be altered by simply switching the style embeddings. Mix-StAGE also allows for style preservation when learning simultaneously from multiple speakers. We also introduce a new dataset, Pose-Audio-Transcript-Style (PATS), designed to study gesture generation and style transfer. Our proposed Mix-StAGE model significantly outperforms the previous state-of-the-art approach for gesture generation and provides a path towards performing gesture style transfer across multiple speakers. Link to code, data, and videos: http://chahuja.com/mix-stage