Alexa Voice features are call and response, but the echo device is also a Bluetooth streaming speaker. That doesn’t require a call and response format. It does, however, require another device to stream to the echo and decide what and when to stream.
Also, I don’t think that option is available just with Amazon voice services, I think you have to have the actual echo hardware, although I’m not sure on that one.
So some community members have use this in the past to play voice announcements on an echo just as you would play a custom MP3 streamed from your phone.
(Note that the following thread mentions landroid – that was the previous version of LANnouncer.)
So it can be done, but there’s no question that the native feature that’s discussed in the verge article would be way easier to use. 