Some bingo sites do require your card details before you can claim their free bingo bonus. But, there are also sites not requiring anything else than signing up with personal information.
However, if you are reluctant to give out any of your details, it's impossible to play free real money bingo games as the sites do need to know who you are. And, in a case where you win money, the bingo site has to have your real details with a confirmed payment method.