Yes, it is absolutely possible to get the cookie from domain1.com by domain2.com. I had the same problem for a social plugin of my social network, and after a day of research I found the solution.
First, on the server side you need to have the following headers:
header("Access-Control-Allow-Origin: http://origin.domain:port");
header("Access-Control-Allow-Credentials: true");
header("Access-Control-Allow-Methods: GET, POST");
header("Access-Control-Allow-Headers: Content-Type, *");
Within the PHP-file you can use $_COOKIE[name]
Second, on the client side:
Within your ajax request you need to include 2 parameters
crossDomain: true
xhrFields: { withCredentials: true }
Example:
type: "get",
url: link,
crossDomain: true,
dataType: 'json',
xhrFields: {
withCredentials: true
}