I have frequently found that when I get an inbox notification for a different site in the network, it does take me to the right place, but I need to log in separately in order to e.g. respond to a comment.
Surely, the site knows I must be logged in before I can see my inbox? Could it use that information to log me in to the sister site when I click on the notification in my inbox?
Sorry if my terminology or tags are off, I'm not sure what all these things are called.