After hours of online research and calls to Verizon groups, I finally found the right rep who told me this feature has not worked since they rolled Fios digital phone service out the door. He could not tell me when it would be fixed or how I would be notified when corrected. The number I contacted was 1-800-837-4966.
Guess what? It still doesn't work. My experience with the landlines call intercept service was the best. The digital option just does not work. They should just drop it as an offering. I am also afraid that its performance varies geographically. Maybe OK some places but not so good in other places.
I see many posts on the Verizon forum complaining that Blocking Anonymous Calls feature doesn't work, and many replies going back years. Here it is now 2015, and it still doesn't work. I have BLOCK ANONYMOUS CALLS turned on, yet I regularly, daily, get calls from PRIVATE NUMBER, UNKNOWN CALLER, V12345678910 (some long number following a V).
Recently read about NoMoRobo which I'm hoping will stop some of these. It's a slightly different slice of unwanted calls in that it's not about anonymous, but about robo calls. I'll give it a try. As for anonymous calls, Verizon FIOS has completely failed to provide this service that they claim is a feature of my plan.
I have the same problem mid March 2016. I get unavailable, 000000000, and many many cities listed for caller ID. Verizon is promising what they are not delivering.