The CLEO signoff is misunderstood by both CLEO's (often deliberately) and by applicants. It is not "permission" to own the weapon; "permission" is granted or denied in the state or local law, or assumed if there is no law against it. Nor is it a background check, although some CLEO's do run a check.
Basically, it is a cop out by BATF, for understandable reasons. They really can't be expected to know about the law in Duckass, Idaho, or to know what the Duckass town council did the night before. The CLEO, by signing, certifies that there is no law in the applicant's jurisdiction that makes ownership of the item illegal. That should be it. Sometimes, it is not.