I've checked all the settings on my J3 and still don't receive alerts for my text messages. I've turned on and off the do not disturb and restarted the phone and still no luck.
You have volume settings, within those settings you can select notifications, system etc... You have settings within the messaging app... You have (if you are running Marshmallow) permissions where you have to allow messaging app to notify you... You can clear the app cache and force stop the app to freshen it up a little.... If all that failed, try wipe cache partition that fixes most maladies....
Welcome to Android Central! Does this only happen when you're in that actual message thread, or does it also happen when you're on the homescreen, or when the phone is asleep?
Are any power-saving modes turned on? The more aggressive modes can prevent apps from pushing notifications while the phone is asleep.