Are simulators owned by airline operators or are they property of private companies who are specialised in this business and offer the services on an outsourcing basis to airlines that require it?
I'm talking about proper simulators like the Airbus or Boeing ones and not PC based ones.
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).
In addition to the good information in @ZachLipton's answer, many aviation training companies offer simulator training for aircraft types other than those that most airlines fly, for example the Kodiak, Citation, or King Air.
This type of training is often used for training individuals flying their own advanced aircraft or for professional pilots. Pilots flying for corporate or charter outfits will typically do initial and recurrent training with such aviation training facilities.
Examples of such training facilities would be the Spokane Turbine Center, Flight Safety International, SIMCOM Aviation Training, and many others.
In addition to these companies that specialize in aviation training for advanced aircraft, many flight schools that specialize in primary flight training also include training in high level FTDs—which have also been used by some airlines, at least in the past. Examples of such schools would be MSU and UND, among many others.
Yes there are private companies that operate Full Flight Simulators and sell simulator time to users, either airlines or self funded students. Pan Am is one example.
Traditional airlines used to acquire their own simulators. However for low cost airlines, pilot training is not core business and a simulator is not an automatic purchase.
There are more and more self funded students appearing in the aviation industry. Pilot training is becoming an issue: there are over 500 new aircraft per year delivered of both B737 and A320. More than 1 per day of each type! Each additional aircraft delivered to an airline (so not a replacement) requires 4 - 5 flight crews. 10 new pilots. 5 captains and 5 F/Os, all trained in a simulator for captaincy and type rating.
To recap: A Level D full flight simulator for an airliner is a 10-20 million dollar device. And there are private companies that arrange bank financing and then sell hours to airlines and self funded students, either with or without instructor services.
As mentioned earlier, most of the major airlines own their own full motion simulators. In addition, major flight schools also have full motion sims as well as companies like Flight Safety International. The OEMs also maintain full motion sims and can lease sim time to smaller carriers or flight crews attempting to become type rated in an aircraft which they sell.
