Who is responsible for ensuring that enlisted personnel are technically trained qualified and afforded the opportunity to attend appropiate professional military education schools?

Answer 1

That all depends on the specific military you have in mind. Some militaries place more of that responsibility at unit level, whereas others do not.

In the US Military, the initial responsibility will fall on the instructors who put those personnel through their occupational specialty training. This may be provided by other military personnel, as well as civilians (both DoD employed and contractors). In some instances of the issuance of new equipment, the manufacturer may send technical representatives to instruct military personnel on how to use that specific piece of equipment.

Upon reporting to their unit, much of the training will be conducted in an "on-the-job" fashion. The chain of command will be responsible for ensuring that the person in question is sent to the appropriate schools, courses, etc., and the instructors of that school or course will be responsible for ensuring that the personnel they train receive the instruction and are trained within the guidelines established by policy.