One possible solution:
In your LAN, you have a folder on some network resource. This folder has subfolders with access limited to person(s) entering new employees, to specific trainers, and to specific supervisors.

In employees registering subfolder you have a workbook with a table of employees in one sheet, and with table of training schedules in second sheet, where is determined which trainings the employee must have;

In every trainer's subfolder, you have a copy of workbook, where trainer can register his/her training sessions, and their results with trainees. The list of current trainees is on hidden sheet, and is read from employees registering workbook by an ODBC query. This table is a source for data validation list for selecting trainees into training registry table(, and probably as source of any additional info about trainees);

In every supervisor's subfolder, you have a copy of another workbook, where supervisor can see training results of his/her trainees. To get this info, the workbook has hidden sheets, which use ODBC queries to read needed info form employee registering table, and from trainings tables for every trainer. The info from those hidden sheets is used to calculate the training reports the supervisor needs.
In case supervisors also have to enter some information, their workbooks get somewhat more complex - in his/her working table the supervisor has to register the trainee at first, and only after that he/she can get any training info about this trainee. And any info he/she is entering, must be entered into same row where he/she registered the trainee;

In case there is a need to get some info back e.g. from trainer's workbooks into employees registering workbook, or from supervisor workbooks to trainers workbooks, etc., this info also must be read using ODBC queries into specific hidden sheets, and then read from there by formulas into visible tables.

With such system, all users can work without affecting other users. Queries are set to be refreshed on opening (optionally after some time interval too), and user can always force data refreshing too. The changed data in any workbook will be available to querying after the workbook is saved (Queries read data from file on network resource. When user opens a workbook and works with it, it happens in user's computer memory only, and all changes to file on resource - e.g. on data server folder - are done when the file is saved).