Wow this is incredible. I've worked a bit in the conversational LLM space and one of the hardest problems we were struggling with was human interruption handling. From the demo it seems like you guys have it down. Can't wait to see where this goes :) BTW I don't think the demo works on mobile, tried it on safari on IOS and got no response.
They're most likely have two "agents" working in tandem to listen and speak, and it seems like the listener takes precedence over the speaker agent but underneath they share the same context window. Programming wise, probably using multithreading and channels architecture, depending on the programming language.
It might ask for permission to use the microphone. If you can't find it, try going to the website's homepage, where you can enter your phone number to receive a call.