Yes, but even though the server doesn’t respond to only you, it is only reactive in nature. All frameworks for websockets seem to be designed with this intent, even though websockets are designed to be full duplex with no restrictions on who sends the first message.
To allow the server to broadcast on its own terms, my best bet so far seem to be to use one of these frameworks in a Comet model. The client-side script would continually and asynchronously request a new message from the twitch channel, and the server would respond only once there is one.