You will need the Reporting permission to access the people query builder.
To save a query first build your report and then select the "Save as.." at the top.
You have multiple options when saving a query, to modify select the dropdown on the Access field within the Visibility section.
Private: Only you and super users will be able to view, run, and edit this query.
Specific People:Only specific people you give access to will be able to view, run, and optionally edit the query. They cannot delete it.
Anyone can view: Anyone with access to the query builder can view and run this query. They cannot edit.
Anyone can edit: Anyone with access to the query builder can view, run, and edit this query. They cannot delete it.
If you select Access level #2 "Specific People" you will be given an additional section to indicate who else will have access to the save query.
To access a previously saved query select the dropdown in the Saved Searches field at the top.