Answer
There isn't a narrow, targeted permission just for reprinting someone else's archived report — being able to reprint any user's archived reports generally requires Administrator-level access, which usually isn't appropriate to hand out broadly to regular end users. A better alternative for the underlying business need is to configure Distribution Groups at the point the report is originally generated/archived, so the report output is automatically made available to the full intended group of users from the start, rather than needing elevated access later just to reprint on someone else's behalf.