I think that its possible they all act independently, but are all focused on generating Autonomy. I don't think that they can directly oppose the will of their shard, but the method in which autonomy is achieved may be a source of difference between avatars, causing conflict.
However, we know that the Shard of Autonomy likes meddling in the affairs of other planets, which Trell is doing, so I wouldn't be surprised if the shard does direct the actions of all avatars.