I worked for a government agency, back in the pre-caller ID days. I handled phone calls that had to be taken care of immediately. Therefore, I only gave out the central reception number, so if I was unavailable, the receptionist could send the caller to my backup. If the receptionist knew I was in, he or she would tell the caller my name and transfer the call. There were a small number of people who had my direct number, who would, of course, know who I was. So, I always answered my phone with "Hello. this is Nutrax For Nerves." No other identifying information.
The state corrections department sent a notice to every police dept. in the US with the number to call to see if someone had a warrant out for arrest due to a parole violation. They put my direct number on the notice. I started getting calls from people checking warrants. I was amazed at how many never stopped to verify that they had called the right number, when I answered with only my name, not my agency or unit identification. Even worse, they would start spouting personal information. "I have John Smith here. His social Security Number is xxx-xx-xxxx. His drivers license number is 1234567. He last known address is..." Even when I'd start yelling "No! Stop! Wrong number! Don't tell me this!" it often took the callers a while to cease their spiel.
What was worse, I couldn't find the correct number to give callers, much less a number to call to get it corrected. The corrections agency only had a few public numbers, that were always answered with "leave a message and we'll get back to you." They never got back to me.
I finally got it resolved a couple of months later when I happened to be chatting with a coworker and mentioned how frustrated I was. "Hey, my husband works for corrections. They are awful about public calls, due to too many cranks and death threats. I'll ask him to check the internal phone book." Coworker returned with the contact number. The person I called was appropriately concerned and apologetic, and gave me the right warrants number, which was a couple of numbers off from mine. Two days later, I was informed that a new notice went out with all sorts of corrections in giant type. Never got another call.