Depends on whether, I as the public, happen to be a criminal or not. I'll assume I'm a law-abiding citizen.

Unbiased.

Quick to act when you are being attacked to save you. Not necessarily by flagging if things aren't clear, but at least getting there and breaking it up.

Gathers evidence as thoroughly as needed, building up a picture of events from all sources. It's funny as hell when a tribunal manages to find absolute proof of a crime after someone has looted a corpse and immediately logged off with the gear, thinking they are

Quick to kill the criminal.

Harsh if needed. Lenient if not. As in, punishment will fit the crime. Accidental offenses may result in less gear being lost than intentional ones etc.

Respected externally (and internally). i.e. their decisions are not constantly being questioned and seen to be dubious.

Respected internally, in that they can get other tribunals to work with them effectively where needed.

Have rp.