Just to let everyone know, if you subscribe on the behalf of someone else using your credit card, they will get sent a confirmation email with YOUR full real name and address. Ed Zotti suggested this method in the announcement, but didn’t mention the confirmation email.
Just FYI, please be careful.
This was corrected on day 2. I decided to re-up for a second year today and the confirmation email doesn’t have your name and address now.
Berkut, thank you for bringing this issue to our notice.
It has since been rectified.
As confirmed by Kat in this thread:
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Well, for anyone still worried about it, I just subscribed myself by credit card, and the confirmation email just said:
So, the name, address, and even the X’ed out CC# aren’t being included in the email to your sponsored Dopers anymore.
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I never understood the problem. Okay, the last 4 digits of the CC isn’t cool. But the real name and address? You have to have that anyway in order to reimburse the nice poster who registered you in the first place.
I just today sent my $5 to the generous poster who already signed me up. Thanks a bunch, NCB!
Even the last 4 is no biggie.
People overreact too often. 
Well, some cc holders are offering gift subscriptions for no money in return. All they need is the username and password of the to-be-sponsored member. In such a case, they might not want their privacy compromised.
And seeing from the generous responses from members to pay for others, we thought suppressing these details would make it easier on our members to sign others up.