If you won a big prize, they would most likely send you paperwork, like an affidavit, to complete and get notarized. In that case, you would need to provide your email address or your home address.
It is important that you do not give out your social security number (unless required on an affidavit). You would not ever need to give your credit card information or bank account information.
I do not think you would be asked for much information over the phone. You should be very careful, and you could always ask for their phone number and call them back which would give you time to verify that they are legit and that you did enter their sweeps.
From my own experience, I have only received a handful of phone calls. One was for an item of jewelry. The sponsor wanted to know if I would accept the prize and wanted to make sure I knew I was responsible for paying taxes. Then, I was sent an affidavit via email.
Another phone call was for a prize pack (T-shirt, baseball hat, sunglasses). The sponsor wanted to confirm my mailing address before actually mailing the prize.
The third was for a hotel package. The sponsor wanted to confirm that I would accept the prize, and sent an affidavit via email.
So you see, there is a lot of variation, depending on how the sponsor chooses to contact you. Most of the time, I have received affidavits via regular mail, or Fedex, without prior phone notice.