Let's bring back the noise, Endofile! Has this been happening since activation? Did you restore backup from previous iPhone? Please complete a soft reset by holding Power and Home keys until device restarts and test again.
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I did all the actions you recommended before submitting the question.
I visited the Verizon Store where I got the phone, and the personnel there where equally stumped and suggested I visit the Genius Bar at an Apple Store.
I did not have to talk with a Genius at Apple. One of the floor personnel was able to confirm that the phone was in fact working correctly. i.e. When pair with an iWatch the audio alert come through the watch.
Case closed. I like the phone. I like the watch. But I don't like the way this feature is set up. Maybe with iOS 9.2 and iWatch 2.2 they will change it.