Integrating robots into populated environments is a complex challenge that requires an understanding of human social dynamics. In this work, we propose to model social motion forecasting in a shared human-robot representation space, which facilitates us to synthesize robot motions that interact with humans in social scenarios despite not observing any robot in the motion training. We develop a transformer-based architecture called ECHO, which operates in the aforementioned shared space to predict the future motions of the agents encountered in social scenarios. Contrary to prior works, we reformulate the social motion problem as the refinement of the predicted individual motions based on the surrounding agents, which facilitates the training while allowing for single-motion forecasting when only one human is in the scene. We evaluate our model in multi-person and human-robot motion forecasting tasks and obtain state-of-the-art performance by a large margin while being efficient and performing in real-time. Additionally, our qualitative results showcase the effectiveness of our approach in generating human-robot interaction behaviors that can be controlled via text commands. Webpage: https://evm7.github.io/ECHO/