Google Home stores everything after the hotword trigger until the light goes off (you stop speaking).

Each action has a card that explains what triggered it, what device, and what result was given, and you can listen to it. You can also turn recording storage off entirely, or delete by device or date ranges.

Honestly the only bad part is it's buried several layers down in the options and account activity, where most people don't go looking. If you do care though the privacy options and controls are pretty good.