Most instructors are freelance in that their time is independent. They could work for several schools for example and they charge any one school/club for the hours flown (depending on the agreement they hold).
The cost of owning and maintaining an aircraft, I would have thought, would be prohibitively expensive for one person to do. Therefore, they would have to join up with other instructors to share the costs. And, whilst they are flying, they would need someone to look after the office, man the phones the etc. And then, they'd end up with a flying club!
I do know that many driving instructors operate through a franchise which is not dissimilar to a flying club!
Cheers
Whirls