As of now, for physical looting each toon has to loot on their own.
You can have skinning on your background toon, with a macro to target last enemy and /use skinning(master) or whatever skill.

In a dungeon or pretty much anything which is not quest specific... as in any one toon could loot it, but then it is gone for others..
You can use the master loot system, with your main as the master looter.
Then just assign the loot to whoever, who receives it without ever having interacted with the corpse.