Screenshots on FaceTime only capture the other person

If you just take a screenshot of your lock screen or home screen, if you go to Recents, are you able to open the complete photo? I just tested in a FaceTime call I was in after my first response to you. I tried both a screenshot and a Live Photo in FaceTime, and in the Live Photo, I do not appear, only the subject does. This is different than a screenshot. Is it possible that you are tapping on the Live Photo button, rather than taking an actual screenshot with your iPhone? Taking a screenshot requires you to use two buttons on the outside of the phone, not on the iPhone screen.

When you take a screenshot while in FaceTime, you should get exactly what it on the screen, which would be the other person and you in a thumbnail view. Are you sure that you are not seeing you at that time? You wouldn't see your thumbnail if you have turned the screen away from you.

So, if you go to the Recents Album on your iPhone, you are not seeing yourself? Have you opened the photo from the album? It seems very strange that you can see yourself when you take a screenshot, but then do not see yourself when you open the screenshot. That could only occur if you edit the photo.