I personally think it has to be a mix of both, having every route have multiple operators is a terrible idea, there wouldn't be enough money to go around, similarly having one operator for an entire region wouldn't work either. They'd have no incentive to make improvements and standards would drop.

I think 'key' locations need to be served by multiple operators, to allow easy travel to different parts of the region without having to purchase mutli-operator tickets. But then multi-operator tickets should be available should you want to visit further afield.  I would define a key location as a town or city that someone from an outside area may want to travel to. These would include places like Newcastle, Durham, Sunderland, Middlesbrough, Darlington, Metrocentre etc. 

Whilst I know it would never happen, in an ideal world those key services would accept any operator ticket to reduce the need for identical services running alongside each other competing for passengers.

So say you want to travel from Crook to Newcastle, you would buy an Arriva Day Ticket on the X46 to Durham, you could then use GNE's X21 to travel to Newcastle. 
Similarly, if you want to travel from Consett to Darlington, you could buy a GNE Day Ticket on the X15, then use the same ticket on the Arriva 7. 

But if you then wanted to travel around the local areas you would have to purchase a multi-operator ticket.