Have you reviewed the active display settings and privacy settings?
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
We definitely want you to be able to receive all your text notifications. Let's check into your settings to make sure everything is set correctly on your Droid Turbo.
To clarify, what texting application are you using? Also, when did the issue with not receiving your text notification when the device is sleep begin?
If you go to Settings>Sound & notification>When device is locked, what is it set to?
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