Hello, pinecone5.
Thank you for using Apple Support Communities.
From your post, we understand you're unable to save FaceTime Live Photos. We'd be glad to help. First make sure that you're both are up to date on software and be sure that both have turned on to allow FaceTime Photos. Check this article for more details: Take a Live Photo in FaceTime on Mac
"FaceTime Live Photo requirements
To capture a Live Photo in a one-to-one FaceTime video call, make sure both you and the other person on the call have any of the following:
A Mac with the latest version of either macOS 10.13.6 or macOS 10.14.4 or later
For more information about the latest versions of macOS, see the Apple Support article Apple security updates.
An iPhone with iOS 12.1.4 or later
An iPad with iPadOS 13 or later
To capture a Live Photo in a Group FaceTime video call, make sure you and the other people on the call have any of the following:
A Mac with macOS 10.15 or later
An iPhone with iOS 13 or later
An iPad with iPadOS 13 or later
This feature is not available in all countries or regions".
The photos should be saved in the Photos app: Browse and view photos in Photos on Mac Check and make sure none of the photos are hiding.
If you're using iCloud Photos, use Safari and sign into iCloud with the Apple ID that has the photos saved to. From here you can click on Photos to see if they uploaded to the cloud.
Are you able to take Live Photos on your iPhone with no issues? Does your mother's Mac have enough storage space? How to free up storage space on your Mac
Keep us updated and we'll go from there.
Cheers!
