Well from Clan to Clan, a challenge is issued.
"I, Star Colonel Amen Jerricho of Clan Coyote's <insert Cluster Name", I challenge you, Ice Hellion surats, for control of X. With what do you defend X?"
Then the Ice Hellions reply, and it usually involves picking a venue for the fight. Once the attacker knows what the defender brings to the table, the attacker decides what force they will commit to the fight. Sometimes the actual bidding and fighting is handled by a subordinate, so in my example above, Star Colonel Jerricho puts it to his Star Captains to bid for the right to attack the target, then observes. At Galaxy level, the same thing would happen between Star Colonels within the Galaxy. It really depends on the size and importance of the Trial.
Between warriors in a unit, it needs to be approved or validated by the unit's commanding officer. So a Trial of Possession for a mech, for example. The commanding officer can refuse to permit the Trial, though I think they need a semi-decent reason.