Do you have the Drivatar designs disabled from the game option menu? If so the designs won’t appear on anyone’s cars, regardless of if they are a Friend or not. I believe you can also toggle the livery option from the pause screen during a race if I remember correctly.
The only reason I bother sharing liveries is so that my Drivatar uses mine instead of the handful of liveries from the same handful of people that we see every single time we play. To my knowledge, my friends have been seeing mine. I know when watching my wife play and watching one friend play on my One, they saw my Drivatar using my livery.
I always get a kick out of seeing Drivatars of people I’ve never met using my liveries. Just recently in the same career race I saw my Nightmare Moon Lexus LF-A, and my Mare Do Well McLaren MP4-12C, both driven by people I didn’t know. One turn on Prague I got the car I was driving between the two other ones, it made a great screenshot.