Just to clarify what Tommyp said...you really can't call the phone company and ask for the private number to show up on your phone. What you can do is request that you not accept blocked numbers so anyone that has their number blocked by default to be forced to dial *82 in order for them to call you. If they don't dial *82 and they try to dial your number then your phone won't even ring and they'll get a message telling them that they need to unblock their number first.
Now there are Tricks that you can do which could reveal a blocked number. Here's what you can do. You can forward your calls to a friend whose phone is handled by a different LEC (Local Exchange Carrier). Have your friend then call the calling party back by dialing *69. Basically your friend is now dialing a long distance call and the number they dial will have to show on their bill.
The other thing you can do is switch your telephone service from unlimited local calling to per call billing (this is good if you don't make a lot of calls to begin with otherwise it could get costly). When you do that then any private call you receive you can then dial *69 to call them back. You won't be able to see the callerID on your phone but you will be able to see it on your phone bill since they bill you by the call.
Casey
EDITED:
How do I defeat Caller-ID blocking? this site has great info on all things telephone including your question. I was not completely accurate in what I stated above. Your friend needs to be in a different LATA not LEC. Looks like multiple LECs can be part of the same LATA which would defeat the *69 feature if they're in the same LATA.