Hi AlexKravchenko,
Screenshots are saved to the desktop by default, so let's have you check there after taking a screenshot in FaceTime. See that information here: Take a screenshot on your Mac
By default, screenshots save to your desktop with the name ”Screen Shot [date] at [time].png.” In macOS Mojave or later, you can change the default location of saved screenshots from the Options menu in the Screenshot app. You can also drag the thumbnail to a folder or document.
If the screenshot isn't saving as expected, test after restarting your Mac, as restarting can help clear up unexpected behavior: Shut down or restart your Mac - Apple Support
Restart your Mac
On your Mac, choose Apple menu > Restart.
Alternatively, you can also take a Live Photo with FaceTime. Check out those details here: Take a Live Photo in FaceTime on Mac
1. In the FaceTime app
on your Mac, choose FaceTime > Settings, then click General.
2. Select the “Allow Live Photos to be captured during Video calls” checkbox.
Kind regards.