Send into the caller's private chat on this worker, opening it if needed.

The chat is addressed by (parent, owner), so a client never has to open one first — and nothing is written until this call, which is what keeps the registry to conversations people actually had rather than surfaces they glanced at.

Same request and response as the worker-scoped send. Nothing identifying the chat comes back, because the caller never needs it: it reads and subscribes through this same parent-scoped surface.