Break News Permissions Up to News & News (super user)
News posted to a school site belongs to that site, and people with News (super user) can edit any news article from any site.
Ex.: a teacher is given News rights. He can post news to a School Site he works in. The Superintendent has News (super user), so she can edit his news as well as her own. The District Clerk has News rights only. That means she can only post news on the District Site, not schools.
Cross Posting rights would only belong to Super Users, simplifying the News Post for non-super user cases.