Hugo vs The General Data Protection regulations (GDPR) in EU (EEA)

Disqus delivers the comments for a post to non disqus logged in users as well.
So they see witch posts from what site are show in a browser. They know e.g. the browser and the IP.

A user (Jane) visits the Blog of anySite1. In order to comment a post she signs in (disqus login). So she has agreed to the disqus terms (when she originally created the disqus account).
Now she is logged into disqus, she adds a comment on the post on anySite1.

Then she spots a URL to mySite2 and surfs to mySite2. Her browser has the disqus cookie set (from anySite1). She instantly sees the comments on mySite2. She may even comment a post on mySite2. I am the admin of mySite2 and the disqus account for mySite2.
I can analyse with the disqus admin panel.
Now the question is: Do I process personal data there - now or in the future? If yes - Jane did newer agree to me (mySite2), that I’m allowed to do that. She did agree to the disqus terms, she may have agreed to the anySite1 terms - but nothing on mySite2.

Am I wrong about this?
May her agreement to the disqus terms be “transferred” to me?
If google analytics needs an agreement from the user (to the site)- why not disqus (there is an admin panel) to the site?