If your messages on your iPhone do not show full preview of the contents of your lock screen, change it.
By default, iPhone with Face ID for Content Previews for Notifications hides until it can be verified. This is not so much because Face ID is super fast and almost immediately. If you have your iPhone on the desktop and in a position where the Face ID is not working, receiving a notification would require you to download your iPhone and verify your face to view the preview.
The good news is that you can always change this to display the content of the message, and it's just a quick switch button in the settings.
Open the Settings app from the Home screen.
Select "Show Previews" at the top.
From now on whenever you receive a message, it will always show preview of the content if your iPhone is locked or unlocked – need not be verified with face ID first.