If this is for Google Messages, does this happen while you're using the phone or all the time? If you're using the phone I think Messages just displays the notification; doesn't vibrate/sound it out.
Also, are you logged into any other session (i.e. Web portal)? Messages is notorious for messing up notifications with multiple sessions open, so another location may be getting the 'full' notification while the rest get only pop ups (or none at all).
Have you checked the category section to make sure that the conversations you're seeing aren't individually set to a different ringtone or silent?
Are you using any smartwatches or notification apps that may be 'intercepting' the notifications?