Each model is given the entire thread, so you can switch anytime. But they don't know who wrote what part and it's not important. The output text is predicted based on the entire contents of the thread and the system prompt says "I" for the LLM and "you" for the user regardless of which model is the "I" at the moment.
They don't "know" anything that isn't submitted as part of the prompt (ie: the context). I haven't heard of any vendor adding "model-aware" info to the system prompt such that they would know when you changed models, nor why that would be helpful.
