I'm certain they get your video data without your consent when the agencies have a warrant. I think it's very likely that they won't necessarily require a warrant, either.

Consider the Nancy Guthrie case. The owner wasn't around to give consent, and the camera didn't even have an active subscription, yet law enforcement was still able to recover video from Google's systems.

The only way it could be as you say is if the video was only stored locally without any remote access, or if the video was encrypted with keys only you control. Google clearly is not doing this. I really, really doubt Amazon is.