The Report Group role will allow someone to share reports across groups they share memberships. However, they can share across all users individually. The Report Global role will allow them to share across all groups. Keep in mind that while they can share across their own and global groups, it also gives them rights to manage groups within each "domain" (their groups and global). Manage is defined as the ability to update, delete, export, and share.
We typically grant either of these 2 roles along with the Publisher and Scheduling role so they can share in those methods as well.
Feel free to find more info here (link is for Istanbul): Reporting roles
