Echo has an always on microphone as well. The story is that audio gets sent to a temporary buffer on a server somewhere to listen for your trigger word, and I trust them, but it would be so easy to start storing that data if they wanted, (or rather, the NSA wanted).
I'm not normally really intense about privacy (relative to other tech people I know), but all these devices creep me out.
No, the processing for the trigger is done locally, no network communication until the trigger is recognized. That's why you can only have one of a couple of trigger words.
The same is true of most modern android phones with Google Now AFAIK. E.g Moto X was one of the first phones with a low power dedicated ASIC for wakeup-word recognition to further make it sustainable on mobile devices.
I was under the impression that the audio is stored locally while listening for the trigger word. Once the trigger word is recognized, Alexa begins recording and sends the result to be processed.
I'm not normally really intense about privacy (relative to other tech people I know), but all these devices creep me out.