The codes don't specify a lot of detail on the loading.
I would assume that vertical handrail loads and vertical live loads on the stairway itself are not combined, but I'm not aware of anything that says that.
On whether both rails are loaded- I would think that would depend on how wide the stairway is, as to whether enough people could congregate to load both railings (or whether that's considered likely, even if possible.)
OSHA has limits on stairway deflection in some cases, but they are large, and for limiting deflection on cable-type guardrails. That may be in the "construction" standards rather than the General Industry standards. (Neither would normally apply to residences).
I was thinking there was a maximum 4" gap provision in those midrails for rails accessible to kids.