Is there a way to hide it?
WhoIs Info for stackoverflow.com
Registrant:
Jeff Atwood
410 Clayton Ave
El Cerrito, California 94530,
United States
Registered through: GoDaddy.com, Inc. (http://www.godaddy.com)
Domain Name: STACKOVERFLOW.COM
Created on: 26-Dec-03
Expires on: 26-Dec-14
Last Updated on: 13-Apr-08
Most of these companies will give out your hidden info at the first request. At least enom does so, I've read somewhere they give out info to any request that mentions the word "infringement". I've also heard there are sites that watch and gather whois updates, inclusive the very first registration data before whois privacy has been activated. They sell then this info to whoever pays.
If you want privacy use fake info. But then you are traceable via payment operations.
So the answer to your question, in the long run that will not be possible.
I don't know how you can do it or how to do it. But I do know that I have my domain through Dreamhost, and they provide a way for me. So when I do WhiIs on my domain, I get this:
geekjuggler.net Private Registrant
geekjuggler.net@proxy.dreamhost.com
DreamHost Web Hosting
417 Associated Rd #324
Brea, CA 92821
US
+1.2139471032Record created on 2007-06-24 14:29:55.
Record expires on 2009-06-24 14:29:55.