Hello Robert,
Thank you for using Apple Support Communities! I understand that when you send a Live Photo via email or Messages on your Mac, a still image rather than the Live Photo is sent. I would be happy to help explore some options to consider.
Depending on the method that you're using to share the Live Photo, it can be normal for the image to be converted to a more compatible format as outlined here:
Using HEIF or HEVC media on Apple devices - Apple Support
iCloud Photos preserves media in its original format, resolution, and frame rate. If your device can't fully view, edit, or duplicate HEIF or HEVC media in iCloud Photos, or displays it at a lower resolution, upgrade to iOS 11 or later or macOS High Sierra or later.
When you share media using other methods, such as AirDrop, Messages, or email, it might be shared in a more compatible format, such as JPEG or H.264, depending on whether the receiving device supports the newer media format."
When I share a Live Photo via email on my Mac, I can confirm that it converts it to JPEG and doesn't retain the Live Photo format. One workaround I suggest is to export the Live Photo in the original file format from the Photos app on your Mac. This would export a copy of both the still image as well as a video file that you can then share by email or text message:
Export photos, videos, and slideshows from Photos on Mac - Apple Support
Depending on the device or person you're sharing the photos with, AirDrop or creating a shared album is another option to consider:
Cheers!