That would be known as cros-site scripting and most browsers
disable that now, as it is a security issue if an application can
read cookies from a different site domain.
Sites that use cross-site data have to either pass it at the
time accessig the other domain (via URL) or use a single database
to record and recall data between applications.
Passport is a good example. Even though you can use Passport
on any site that offers it, the site ultimately tranfers you to the
Passport website momentarily to collect your login and then
transfers you back to your site along wth the credentials in a URL
variable so your own site can then record the cookie and state you
are logged in. It doesn't actually read the Passport cookie from
your own site.
If you were to create a DB that applies to multiple sites,
you could figure out a way to populate session variables on
separate sites by quering the DB for the data if there is no data
currently stored, or if it detects that the referer was a different
URL prior to loading the current site. Once it queries the data it
can store the data in a session variable.