I'd agree aside from medical reports... I think med forms should be under lock and key at all times. Available only to Company Management...
I guess this is easier if you have a full-time company manager and your rehearsals take place during the regular business hours.
Yes, this is one of the differences between dance and theater. Though most theaters have company managers, most times they are not as present/ involved as dance company managers.
In most larger dance companies all medical info is handled exclusively by company management, in theater SM is usually the first point of contact and passes info/ responsibility on to the CM.
I think it is very important for ASMs to be informed about medical conditions of the cast. Often the ASM is point of contact backstage. If (for example) a cast member has diabetes or some other chronic illness, I as the ASM need to know this. If the SM is calling the show and the actor starts to get sick, the ASM is the first person to help that cast member. It is my responsibility to treat this information carefully, but it is also my responsibility to know it.
It is your responsibilty as the PSM to make sure your staff is informed and aware of procedure, but an uninformed staff can be a dangerous staff.
Of course we need to be discreet with our cast's information. This applies to contact info, overtime pay, hazard pay, medical info, anything that we as stage managers end up knowing.