Try not to make the training requirement and documentation harder than it has to be.
1. You have your "job descriptions". Sit down with appropriate managers and list the basic requirements and skills. Decide which are a pre-requisite for getting the job and which can be trained for.
2. Use this as a checklist for training documentation along with your list of items trained for during orientation.
3. Those skills that the employee has when they come to you can be so noted by the manager that made the decision to hire him/her.
3. As each job or skill is mastered, the employee and the qualified trainer initial or sign off and date that item.
4. Keep a list of employees "in training" and review where they are in their training at some specified interval. This should be done by managers the employee reports to along with whomever is in charge of your training program.
5. Make sure that you have your procedure documented (but not overdocumented).
This works well and is very easy to live with. If your documentation is recorded in a computer program, you should be able to generate your check-off sheets easily. If your training documentation is done only on paper, all you need to do is copy you "job description" and use it as the check-off. Each manager is responsible for assessing only those jobs under his authority, yet the employee's training file will contain documentation of all the jobs or functions s/he is qualified to perform.
Believe me, the corporation I work for understands multi-tasking and cross training! I have done everything from loading trucks, maintenance work, computer and network installation and administration, production, and QC/QA testing and inspection, auditor, training coordinator, safety program administrator and janitor, all while being the quality system manager.
In over ten years of audits including ISO and QS we never had a nonconformance directly against training. We did, internally identify some training issues as root cause of other nonconformances, but hey, that's why you audit - continuous improvement.