Not getting notifications when texts are sent??

See this for information on how Notifications work when you have an Apple Watch. Notifications on your Apple Watch - Apple Support When you say Sent, I'm guessing here that you are referring to messages sent to you, is that correct?

Read the footnotes on this webpage about using your Apple Watch when your iPhone is not nearby

https://support.apple.com/en-us/HT205547

1. To receive SMS, MMS, or push notifications from third-party apps on your cellular Apple Watch, your paired iPhone must be powered on and connected to Wi-Fi or cellular, but it doesn't need to be nearby. You also need to be signed in to iMessage on your iPhone.