However, I can not detect the cookie at domain2.com.
You aren't going to be able to nor should you be able to. It's a security feature of the browsers.
A solution would be to just make a redirect to the other domain where the
cookie is set and then return.
That would be one solution.
Question is: Can I not set a cookie at domain1.com to work at domain2.com?
No, you can't. You could, however, if you were trying to set it for a different sub-domain (eg. subdomain1.domain.com and subdomain2.domain.com).
thnx,
Chris