I’m having the same problem. After much testing, I have discovered that the alert tone DOESN’T ring when the Messages app is OPEN (as it would be if you’re having a back-and-forth text conversation with someone), but it DOES ring if the app is CLOSED. I think the assumption is that if the app is OPEN and in use, then you don’t need an alert to ring each time the other person sends a reply. While that might seem logical, it really is a false assumption. Sometimes there are significant delays between sending a text and receiving a reply, and the app might be intentionally left open so that you can read a very important reply as soon as you receive it. The irony is that the opposite often happens—you start working on another device while waiting for the alert to tell you that you've received the very important reply, but the alert never rings. So instead of immediately reading the reply (did I mention that it might be very important?), you won’t even know it’s there until you finally check the Messages app to see what’s holding it up.
I’m still trying to find a way to make the alert tone ring each time a message is received, even if the Messages app is open. I’ve tried all the suggestions I found in various discussions, but none work (and turning Bluetooth off wouldn’t be a good solution even if it did solve the problem, because it needs to left on for many other apps such as “Find My,” which won’t be of any help when you need it if it’s been disabled). If I find something that works, I’ll post it here.