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I'm not sure you can get this from the REST API. Quickly looking, I don't see it coming back from a person or streams query.
What you can do though is make an AJAX request to /index.jspa from your application. Take a look at the Location header in the response - that will have the streamID in the URL. Using that ID, you can make v3 REST API calls to get information about the stream.
Cool idea, thanks! )