No, they are not self-dependent per se. They all have their own separate attributes, but they still have a spark of Vishnu within them. These would be Ganesha, Lakshmi, Murugan, the Alwars, etc. I would say that this is akin to Vishishtadvaita.
Wheres Shiva, and others like Indra, Agni, and the Avatars, are directly emanated from Narayana and are him in different modes.
Some are direct emanations and are Vishnu himself, while others are separate entities with a spark of Narayana within. Nevertheless, all are connected with Vishnu in one form or another.