GoDaddy Requires Users to Compromise Privacy to Transfer Their Own Domains
godaddy.com
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So you have to the transfer get it rejected, THEN turn off privacy, then do it all again.
On the other hand, that just makes it that much sweeter once the domain finally lands at Route53 (etc) and you no longer have to play Godaddy Renewal Roulette and wonder if your next renewal will be 50% higher than normal.
(Oh, and BTW, I was happy to see that if your domain registration is a company, Route53 does quasi-private registraiton at no extra cost.)
For any user, its pointless to buy GoDaddy privacy protection if they ever plan to move their domain out because eventually their data is going to end up in public domain since they can't transfer without disabling privacy.