I'm running IOS 16.1 and I'm not hearing any sounds when a text message is received if my screen is locked. My connected apple watch is working fine. There are no messages (banners) coming to the locked screen either.
iPhone 8
If your iPhone screen is locked, you would be receiving notifications on your Apple Watch. See this for an explanation of how notifications work when you have an Apple Watch. Notifications on your Apple Watch - Apple Support
If your iPhone screen is locked, you would be receiving notifications on your Apple Watch. See this for an explanation of how notifications work when you have an Apple Watch. Notifications on your Apple Watch - Apple Support
No sound on text messages being received when iphone screen is locked
