Few days I tried to move my GoDaddy domain to Gandi.net, and GoDaddy denied my request because apparently the domain is registered privately. Why should I expose my name just to transfer?
You should have asked yourself that question before you registered at Godaddy. If you do not want your name to be associated with your domain, you are now locked in forever. To be honest, I don't even know if other registrars do it any different. (I only know that Nearlyfreespeech.net does not lock you in that way, but not about others like namecheap.)
Gandi.net does not offer full whois privacy by the way, so your name will always be revealed.
Privacy can be expensive. If you are in the US, you could try getting a PO box. Post offices around me have them for as low as $42/year. That is about three years of domain registration, but if you have multiple domains or want to keep your home address off the Internet, the cost can be worth it. Plus, now you have a different address for things and it's a locked box that's difficult to break into.
