We make pretty much everything in house, and everything done in house is done by students. We take a lot of pride in our manufacturing, simply because it has always been a strong point and have developed a bit of a reputation for build quality.
Training is simple and experience based, though we do have reading material for CYA purposes. People are considered trained when they are proficient at using the machine safely and accurately as well as able to teach another student how to use the machine. Trained students are then checked off by a mentor, again for CYA.
We have about 12 people on the manufacturing team. Going into a season we shoot for the following, though higher numbers are better

Everyone checked off on the layout tools, band saws, drills/drill presses.
4 checked off on the mill/advanced layout.
4 checked off on the lathe.
Training on the mill and lathe as well as advanced layout techniques are done through a class that the local community college offers each year for FRC students.
Quite a few students on our team take the time in the summer to explore tools that we do not have in house, such as 3D printing and CNC machining.