All Southwest training is done by Southwest pilots, directly employed by Southwest Airlines, in a world class training facility.

Some events are conducted by Flight Crew Training Instructors (FCTIs), some by SWAPA pilots, and some by check airmen (also SWAPA pilots).

Full motion simulator training (not checking) is scoped to FCTIs. I might not have the language of this totally correct but this is what it is basically.

All of the above is contractual minutiae from the perspective of a new hire...the training is very good (I said training, not the schedule of which sucks) and you'll have no idea what group your instructor falls into unless you ask. You have to be a major screw up to not succeed.