If the iPhone screen is on, text notifications go to your iPhone. If the iPhone screen is off, notifications go to your Apple Watch. They do not go to both. You most assuredly can get text notifications on your watch with sounds or haptics, which should alert you to an incoming message. So, I'm not sure why you are missing messages, when either your phone or watch are capable of notifying you.
Read this link about notifications on Apple Watch --> Notifications on your Apple Watch - Apple Support
Personally, I use Haptics on my watch and have them set to Prominent. I get a tap, which I can't miss on my wrist when a notification arrives.