I’m using Zwift on an iMac together with the iOS Companion app. I’ve set my image upload preferences to “Only mine” in the Zwift settings.
After completing a ride, you can select which images will be uploaded. When disabling the HUD (by pressing the little graph icon), the HUD and other on-screen data should be hidden. The HUD is hidden, but if any riders are nearby, their names will still be shown in the image. See attached image.
The rider names should also be hidden, if the HUD icon is disabled.