I think we need to ask several separate questions before knowing the answer.
Is the winning notification legit? Did you receive it via email, phone or regular mail? Can you verify the sender (for email you can right click to see details, for example).
Is there a contact person?
Did you enter the sweeps? What is the full, correct title? Can you find it on SA, or via Google?
Providing your ssn for tax purposes could be legit, and members have been asked, even for books and t-shirt wins. You are doing the right thing by asking and verifying...always be cautious.
About Woman's Day, I won a $60 handbag several years ago, but it just showed uo in the mail. That doesn't mean all their prizes are sent that way.