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Sounds like the HD voice feature to me. You can test this by contacting customer service (*611 then send from VZW phone or 1-800-922-0204) and asking for them to add the Decline HD voice/advanced calling feature on the lines. Then call each other and see if the issue persists. Of course there is a number of other things that could cause this, but based on your description I believe it has to do with this feature.
Let me know if you try this 