Depends on the Fallen. Anduriel has generally been content to work with Nicodemus. We have no idea on the preferences of Thorned Namshiel. It might also depend on the "quality" of the host.
Two points:
(1) All of the Denarians who are partners (like Nicodemus, Tessa, and Deirdre) are called by the host's name. Those who were fully subjugated (like Magog and Ursiel) are called by the fallen's name. In Small Favor, Thorned Namshiel was called Thorned Namshiel.
(Counterpoint: In Skin Game, Ursiel's host seemed to be in charge, so the same fallen might subjugate one host but "partner" with another. Or start out "partners" with a long slow slide into subjugation.)
(2) There are really strong indications in Skin Game (Michael's conversation with Nicodemus in the Vault) that Nicodemus isn't actually an equal partner -- that he's not a subjugated slave, but that he's still manipulated by the fallen and after thousands of years of such manipulation he's no more free than Ursiel's first host was. He just doesn't realize it.