Mostly this post is for myself, but others can benefit from it too. Here are the loot types from each boss.

Hel Ra Citadel:

Boss 1 (Ra'Kotu):
-Chests
-Weapons/Shields

Boss Top (Yokeda Kai):
-Boots
-Pants

Boss Bottom (Yokeda Rok'Dun):
-Belts
-Shoulders
-Gloves

Final Boss (The Warrior):
-Necklace
-Rings
(And all other loot types too at least in Normal)

Aetherian Archive:

Boss 1 (Lightning Storm Atronach):
-Shoulders
-Pants
-Boots

Boss 2 (Foundation Stone Atronach):
-Belts
-Hats
-Gloves

Boss 3 (Varlariel):
-Weapons
-Chests

Final Boss (The Mage):
-Necklace
-Rings
(And all other loot types too at least in Normal)

Sanctum Ophidia:

Boss 1 (Possessed Mantikora):
-Hats
-Chests

Boss 2 (Stonebreaker):
-Gloves
-Belts

Boss 3 (Ozara):
-Pants
-Shoulders

Final Boss (The Serpent):
-Necklace
-Rings
-Weapons


It came to my attention that most people are looking for Helmets and Shoulders for the sets from these trials. In the case of VO and IA, you can farm the first boss in SO to get Hats, and the first boss in AA to get Shoulders. (You can farm vAA first boss with a few people, it just takes a long time but worth the increased chance for VO and IA).
Other people are looking for weapons, and have already started farming Hel Ra first boss for them.

If there is any incorrect information here, please let me know. Or anything else I missed. I'd like this to be accurate.