Amazon has said on numerous occasions that no data transfer occurs without a trigger word hitting the mic -- a feature that was a main point when discussing the safety of having an always-on internet-connected mic in the house.
As for whether or not they're telling the truth, I don't know; but trigger-words have always been a feature that Amazon loved mentioning from a security/privacy standpoint.
Luckily it's possible to check [0]! Although it gets a bit more complicated and can change, my understanding is that currently most people observe it increase it's network usage after it's trigger phrase, but not at other times (it uses the network for other stuff too, but audio data is typically rather large in comparison).
[0] https://www.iot-tests.org/2017/06/careless-whisper-does-amaz...
[1] 10.1007/s00779-018-1174-x <- Might want to use sci-hub
What you should be asking with any "smart" device is "can I prove this device will do no harm to me".
Honestly I have never understood the value proposition of any smart device. Why would I want any of that functionality? Never once in my life have I ever wanted to talk to my TV. I'm beginning to (again) question the wisdom of carrying a smartphone.
Conversely, if they were not doing that kind of thing, they could have removed that ambiguity from what they said.
http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&Sect2=H...
In case the link doesn't work, it's US Patent 9922646.
http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=H...