How does jet training work for private pilots? Are there jet trainers?
There is no FAA requirement for any specific pilot certificate level beyond Private to get a type rating in any jet. In fact, many owner/operators of small business jets (CJ, 500s, Phenom, etc) only have a Private certificates. 

See 14 CFR 61.31

The only FAA requirements are that you have a Private certificate with multiengine and instrument ratings. Additionally, a High Altitude Endorsement is required to take the actual type checkride unless you meet the grandfather requirements of the HAE. RVSM is NOT required to take the checkride, but it is required before flight into or through RVSM airspace. 

You can enroll in a simulator based Part 142 course (think Flight Safety, Simcom, LOFT) and complete your entire training in a Level C or D full flight simulator, but once you complete your training you will have an SOE limitation on your type rating. This is per 14 CFR 61.64. This regulation says you must have 25 hours of Supervised Operational Experience in the aircraft before you can act as PIC in that aircraft unless you meet specific requirements including total time, time in multi-engine turbine (including turboprop) aircraft, military training, etc. 

You can enroll in a part 61 course and complete your training in the actual aircraft and once you pass the type rating, you would not have the SOE limitation under 14 CFR 61.64 since you took the actual checkride in the aircraft as opposed to an FFS. 

You can enroll in a Part 142 initial hybrid program that offers simulator training for the first part and when you have complete the sim portionof your training they move you to the aircraft for the final training segments and checkride. This also eliminates the 61.64 limitiation on your certificate.

Three things to remember: First, ALL type ratings are done to ATP standards regardless of certificate level and Second, SOE or not, insurance companies rule on how much time you will need to fly with a mentor pilot before going out on your own and finally an in-aircraft checkride for your type rating will be far easier than an in in-sim checkride.