Hi there, grandmaEarle,
Thanks for visiting the Apple Support Communities. I understand that Live Photos you take in FaceTime aren't being saved on your iPhone and iPad. I'm happy to help!
In order to save a Live Photo from FaceTime, both your device and the people you are talking to's device(s) need to be running iOS 12.1.4 or later, or iPadOS, according to the below information from Use FaceTime with your iPhone, iPad, or iPod touch - Apple Support:
"To save FaceTime Live Photos, all participants need iOS 12.1.4 or later, or iPadOS. Not all features of FaceTime Live Photos are available with earlier versions of iOS. FaceTime Live Photos doesn’t work with iOS 11."
If the other person is using a Mac, they must be running macOS 10.13.6 or macOS Mojave 10.14.4 or later, and have Live Photos enabled in their FaceTime settings: Take a Live Photo in FaceTime on Mac - Apple Support.
If this does not address your issue, don't hesitate to respond back!
Best!