Create clear API to observe who's turn it is.

_↴ Ported from [flutter/genui#847](https://github.com/flutter/genui/issues/847) — originally opened by [polina-c](https://github.com/polina-c) on 2026-04-01._
_Original labels: P2, front-line-handled, a: api-simplicity_

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Define observable, explicit and easy to discover way to detect if it is agent's or user's turn to say something in conversation.

Details: https://github.com/flutter/genui/pull/846#discussion_r3019205909

Review the linked discussion in the original PR to understand the context. Look for existing conversation state management in the codebase, likely in a module handling agent/user turns. Identify where turn-taking logic is currently implemented and design an observable API (e.g., using an event emitter or reactive stream) that exposes whose turn it is. Ensure the API is explicit and easy to discover, then write tests to verify the observable behavior.

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