Why wouldn't the browser send cookies?
Site A (http://www.sitea.com) sets a cookie for the user.
User navigates to site B (http://www.siteb.com). Site B features integration with site A - click here to do something on site A! The users clicks "here".
As far as the browser can tell, the user is making a conscious decision to make a request to site A, so it handles it the same way it would handle any request to site A, and that includes sending site A cookies in the request to site A.
Edit: I think the main issue here is that you think there is a distinction between authentication cookies and other cookies. Cookies can be used to store anything - user preferences, your last high score, or a session token. The browser has no idea what each cookie is used for. I want my cookies to always be available to the site that set them, and I want the site to make sure that it takes the necessary precautions.
Or are you saying that if you search yahoo for "gmail", and then click on the link that takes you to http://mail.google.com, you shouldn't be logged in, even if you told gmail to keep you logged in, because you clicked on the link from another site?