I'm a quality manager and often employ this type of service.
I've not started one from scratch, but can help with what I know.
1.) In my experience, when a gauge or tool is required, the customer (me) provides that along with instructions.
2.) The third party sorting company can, and often do, make their own instructions, which the customer would have to agree to or provide alternative instruction.
3.) A third party containment company is often ideal to employ, because in the event of a quality failure the second party may need to shut a line down if they have to sort themselves. this could cost the first party(me) a lot of money; we're talking millions of dollars. Often the first party doesn't have resources to continue running their own processes and send their own personnel to the second party to sort product. So we call in the third party.
4.) generally speaking, the third party is one that is approved by the second party, but is paid and employed by the first party. its all a bit gray in that aspect. the third party has an agreement, which is signed and approved by the first party, usually covering liability and other legal stuff.