Past work has demonstrated that autonomous vehicles can drive more safely if they communicate with one another than if they do not. However, their communication has often not been human-understandable. Using natural language as a vehicle-to-vehicle (V2V) communication protocol offers the potential for autonomous vehicles to drive cooperatively not only with each other but also with human drivers. In this work, we propose a suite of traffic tasks in autonomous driving where vehicles in a traffic scenario need to communicate in natural language to facilitate coordination in order to avoid an imminent collision and/or support efficient traffic flow. To this end, this paper introduces a novel method, LLM+Debrief, to learn a message generation and high-level decision-making policy for autonomous vehicles through multi-agent discussion. To evaluate LLM agents for driving, we developed a gym-like simulation environment that contains a range of driving scenarios. Our experimental results demonstrate that LLM+Debrief is more effective at generating meaningful and human-understandable natural language messages to facilitate cooperation and coordination than a zero-shot LLM agent. Our code and demo videos are available at https://talking-vehicles.github.io/.