This is very alarming! We want to make sure that all of the sounds on your phone work awesome. If you go into your settings under sounds is your ringer turned up? If you change the alert sounds there does that help at all?
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If the overall volume is up and you don't have the phone on silent/vibrate, try shutting it off completely and then restarting it. A quick restart cures a multitude of minor problems.
I had the same problem. What worked for me was to go into settings>notifications >messages>Allow notifications. I turned this off then turned the phone off then turned the phone back on and turned notifications back on. It started working again