I can go into sounds and change the text tones and I can hear the sounds but not when I get an actual notification. I have vibrate on and the phone does vibrate and the notification does appear as a alert.
Try going to Settings - Notifications - Messages and make sure that "Allow Notifications" is enabled, make sure that the correct alert tone is set next to "Notification Sound" and then scroll down to make sure that a checkmark is next to "Show Alerts from Everyone". Finally, make sure the mute switch isn't enabled on the side of the device and the "Do Not Disturb" feature isn't enabled.
I have verified Settings - Notifications - Message - Allow Notifications is enabled. If I select a "Notification Sound", I can hear that tone. I have verified there is a check mark next to "Show Alerts from Everyone". The "Do Not DIsturb" is not enabled. I don't get a vibration if "Do Not Disturb" is enable. Ahhhhh the "Mute Switch" located on the side of the phone was set to "Mute". That resolved the problem. I have an OtterBox and didn't see this setting until I removed the Otter Box