Most cargo companies don't have approval for passenger operations, so they can't transport passengers - paying or not.
Technically "passengers" on board of cargo aircraft are not passengers. They are usually called supernumerary (or some other term) crew members. If you are transporting animals and need some people to take care of them (grooms, vets, etc.) they are doing their job on board the aircraft - taking care of cargo. Ergo they are not passenger, but crew members.
Certifying people to be supernumerary crew depends on the company - can be as simple as on-spot briefing by the captain.