Are jet simulators a commercial business or are they owned by airlines?
Both!

Some airlines operate simulator facilities. For example, United has a training facility in Denver with a number of simulators. In turn, they rent out training services to others who will pay to use the facilities (more info on their offerings here). They can also provide trainers, cabin crew training, ground school training, and other such services. Many airlines have similar programs. They've even auctioned off simulator time to customers willing to bid frequent flyer miles for the privilege. 

When an airline doesn't own and operate the necessary equipment itself (even an airline that runs its own simulators may not have a suitable quantity for every aircraft in its fleet), it can contract for such services from someone else. That someone else might be another airline, a company that provides simulators (example) and may offer other training services like instructors, or a provider affiliated with the manufacturer (example).