Well, I can think of one kludgy way it could be done theoretically...
Say on mysite.com you have a page with an image loaded from
myothersite.com. Only the image coming from myothersite.com
is actually a CGI script that reads the cookie for myothersite.com,
and with some clever backend communication, sends the cookie
data to mysite.com. Of course, this wouldn't work if the person
enabled the "Accept only cookies that get sent back to the originating server"
option in Netscape. So considering it may not be very reliable to
do it this way, I would just find a non-cookie solution for
whatever you are doing.