EOM tokens come at the end of every response that isn't maximum length. The other LLM will respond to that response, and end it with an EOM token. That is what is going on in the above example. LLM1: Goodbye<EOM> LLM2: Bye<EOM> LLM1:See you later<EOM> and so on.

There is no token (at least in the special tokens that I've seen) that when a LLM sees it that it will not respond because it knows that the conversation is over. You cannot have the last word with a chat bot, it will always reply to you. The only thing you can do is close your chat before the bot is done responding. Obviously this can't be done when two chat bots are talking to each other.

You don't need a token for that, necessarily. E.g. if it is a model trained to use tools (function calls etc), you can tell it that it has a tool that can be used to end the conversation.