You will need the Reporting permission to access the people query builder.
Saving a query
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 saved query.
To access a previously saved query select the dropdown in the Saved Searches field at the top.
Editing a saved search
Go to the "Saved Searches" dropdown and click on the desired saved search. Then select the "cog" icon, and pick "Edit." Make the appropriate changes and choose "Save."
Deleting a saved search
Go to the "Saved Searches" dropdown and click on the desired saved search. Then select the "cog" icon, and click on Delete. Choose "Yes."