FaceTime photos missing from Photos app on MacBook Pro

Thank you kindly for your reply, and for trying those options. Sorry that hasn't helped.

Re: "The photos are nowhere to be found. I don't think they ever got saved to the computer".

Trouble-shooting:

• You might want to check whether or not FaceTime Live Photos is turned on.
• The person you took the photos of would need to have it turned on too, to allow that.
• Apple has a few system requirements that need to be met too.
• Also: This feature isn’t available in all countries or regions.

Take a Live Photo in FaceTime on Mac - Apple Support

Eg: says:

"FaceTime Live Photo requirements

To capture a Live Photo in a one on 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

An iPhone with iOS 12.1.4 or later

An iPad with iPadOS 13 or later"

Also says:

Set up FaceTime for Live Photos

1. Go to the FaceTime app on your Mac.
2. Choose FaceTime > Settings, then click FaceTime.
3. Select the FaceTime Live Photos tick box. (This option is unavailable during a call.)
4. Note: When you select this option, you’re also allowing others to take Live Photos of you.

Take a Live Photo

Before taking a Live Photo, make sure you’ve opened the Photos app  at least once, so Live Photos can automatically be saved in your Photos library.

(The Photos app doesn’t need to be open when you take a Live Photo.)

1. Go to the FaceTime app on your Mac.
2. Do one of the following during a video call:

• • In a one on one call: Select the FaceTime window.
  • In a group call: Double-click the tile of the person you want to take a Live Photo of.

3 Click the white circular button.

4 When you take a Live Photo of someone, or someone takes a Live Photo of you, you both receive a notification.

Note: If you can’t take a Live Photo of someone, make sure both of you have set up FaceTime for Live Photos."

Re: "FaceTime photos missing from Photos app on MacBook Pro"

My Mac is using an earlier MacOS version than yours.

This quick method works on my Mac ok ... you might want to try it.

Open the Photos App

Click Library (in the sidebar)

Click All Photos (along the top of the Photos window)

Type FaceTime in the search field

(If needed: Click the magnifying glass to open the search field)

(If needed: You could also try searching by the date instead of for "Facetime")

___________

Another way is to Click on Library, and then on Live Photos (also in the sidebar)

More info: Take a Live Photo during a FaceTime call on iPhone - Apple Support