I’m not familiar with Ring, but if it’s anything like their Blink cameras I wouldn’t count on it. The Blink cameras advertise a local only mode, but the functionality is reduced to make it unusable. You lose thumbnails for videos and they take away the ability to bulk delete (IIRC). Have fun navigating into every video, waiting for it to start playing, and then deleting it.

The other catch is if “the cloud” is needed to generate thumbnails, it’s not a local only mode.