Control chatkit session on MCP server

I have an MCP (sse) server configured in the Agent Builder Agent.
How can I get information from the MCP about the user or session that initiated the conversation?

I may not be completely following your question but the ChatKit SDK allows you to set variables that the agent then gets populated. When you go to generate the client secret on the server you add them to the payload. This is in the ChatKit docs.

https://platform.openai.com/docs/guides/chatkit