Click Edit on Saved Search > Notice: That is a special search that returns unrestricted results, even when run from restricted roles. It may only be edited by admins

The notice is caused by the Run Unrestricted box marked under the Results tab of the affected Saved Search.

Notice:

"That is a special search that returns unrestricted results, even when run from restricted roles. It may only be edited by admins".

If this box is enabled, even the owner (who is not an Administrator) could not edit the Saved Search.

The Run Unrestricted box allows the users with non-Administrator role to see the search results but not to edit.