With cURL, I don't think it is possible. I think I can safely say that the majority of sites use PHP Sessions for logging in, etc. These sessions involve storing a cookie on the clients computer. I do believe that it would be possible to get and set the cookie via cURL, but there is no way to make it accessible to siteb, and it would therefore be useless.
One more note: when storing your user's passwords in the database, you must encrypt them! Unfortunately, you couldn't use a one-way salted hash because you need to be able to take the passwords back out. I would essentially use the same concepts as the uniquely salted password method, but use the two way AES-256 encryption rather than Blowfish or SHA.