Yes
Apart from actual phone calls which ring on both by default, the Sounds and haptic notifications and banners for messages etc follow these rules…
If the phone is in use (open) then you get tangible notifications on the phone only. Messages and Mail still appear silently on the Watch in the relevant app without banner..
If the phone is locked you get tangible notifications on the Watch only. If you have notifications set for iPhone Lock Screen or Notification center they still go to the phone, but silently without banner.
If you go into each notification type and use Custom and then disallow notifications the alerts and banners go to the phone always and not to the Watch
I think that may be your option and often allows you do set a purchased ringtone (crazy frog?) fir some apps like Messages.
Notifications on your Apple Watch - Apple Support