The original idea was that the ICs would take turns as the scrum master. I'm sure someone somewhere has actually done that, but it's unusual.
I can understand why the idea of hiring someone to handle that role would be appealing, but it really isn't a full-time job.
Plus, the last thing that BigCo engineering teams need is more politics; they have more than enough of that between the overlapping layers of engineering, project, and product managers.