The best gift cards I've found are the *Simon Giftcards* as they never expire and never charge a monthly fee. They cost $3.00 per card (in addition to whatever money you want put on the card) Most other prepaids charge monthly fees.
Around a third of the major (in-door) shopping malls in the USA are owned by Simon, and they sell these Gift cards at the mall help desk. That's what I always try to get. Otherwise the monthly fees of other cards can eat you alive.
Prepaid credit cards/giftcards can be used in SOME online transactions. The Simon Giftcards usually (but not always) require inputting the card's CVC code for an online purchase, otherwise they will usually decline the payment if a site does not ask for the CVC code. The CVC code may be on the front of the card - with a different 3 or 4-digit code on the card's backside - so it can be confusing. Try both to see which works. A fake address is usually accepted (you just can't leave the boxes blank).
It seems that there is no hard rule for determining if a site will accept a prepaid/gift card. Just try it and see. Sites can often change their creditcard processor (and a giftcard can itself change its payment processor) so a particular giftcard's acceptance at one particular site may be different from year to year. The most recent purchase I made with a Simon Giftcard (AMX) was with Blocknews last year, and it worked then but I can't say if it will still work today. My suggestion is BUY THE CARD and then see who will accept it. My own experience has been that most usenet providers will, and I've used giftcards on many USPs over the years (probably more than I can easily count when I include free trials).
Regarding the 'prepaid debit cards' (that are not sold as 'giftcards') -- It seems some don't *require* a Social Security number -- for non-citizens at least. But it's best to avoid these cards generally since they must be **activated** with a name/address/DOB after you buy them before they can be used and in some cases the card you buy is just a *dummy card* which can't actually be used - so you must wait for the *real* card to arrive in the mail.
As far as a usenet provider keeping your personal details, we should assume that they will keep your name, address, phone, and email address you give them. Your credit card number may or may not be kept, and if it is, it might be on file with a 3rd-party payment processor only, I really don't know.