In regards to the Sox, I've been told and have read that JR is the majority owner. I've seen different figures anywhere from 5% of the club to 21%.
The Sox have so many owners though, that even that smaller amount is enough to claim title. Also in his contract he specifically has been given the power to run the day to day operations of the franchise without having to consult the other owners and or board members. If he decided this off season, "I'm spending 200 million on payroll and am going to go out and sign every free agent all-star I can get my hands on," he has the power to do so.
I assume that only in the case of the usual business standard clauses (i.e. doing something illegal, defrauding the organization, acting like he has lost competency) can he be removed for his authority.