Reading the above posts, for the first time in my life I've thought of the One Ring as the Tolkien version of a femme fatale. But I need to ponder this association further before it forms into a solid thought. (But if I was the type to write fan fics... the history of ME would be rewritten now starting with the forging of the Ring...
)
As for the interaction between Sauron and Gollum - I think at least on Sauron's part, he felt no rivalry from Gollum - or from "Baggins". Gollum was simply too weak, physically and mentally, to appear as a threatening rival. He was nothing, a fly to swat. Baggins, yet an unknown entity, didn't appear too sophisticated either, given Sauron's limited knowledge of hobbits and contempt towards lesser beings. He could, and did, envision Aragorn as the Ringbearer, because he sensed a rival power in Aragorn and refused to believe that a Ringbearer would refuse to use the Ring for power. And the only power that Gollum had, and would ever have, is to be able to catch more fishes. Not remotely threatening, from Sauron's perspective.
I agree with
Inzil about Gollum. I think that more than ever before he just wanted to "elope" from the world with his Preciousss when he saw Sauron's dominance over him in everything, including their claims to the Ring.
I think one of the most important aspects of many of the Ringbearer interactions is the question of trust between the Ringbearers. Since I still can't wipe the femme fatale thought off my mind, the first analogy that came to me is someone meeting their significant other's ex. Knowing that this person held the Ring before you did (or, for the other person, seeing your Ring in someone else's possession), can put a wedge of Ring-jealousy in the relationship between these two people. And you can see that in Frodo's interactions with Bilbo and Sam, when one of them becomes blinded by that feeling and forget the context of their friendship and the current situation. It has always fascinated me that all three hobbits had enough strength of mind to overcome that feeling and to go back,
completely and unsuspiciously, to their previous relationship after such an event. I find it a wonder that not once did a suspicion that "my friend is after the Ring" come crawling into their relationships. Perhaps that's one of the things Sauron underestimated most in the hobbits when he relied on the Ring's guile to mess up its other Bearers. He expected them to fight over the Ring to the death, not to help each other destroy it.
EDIT: xed with Pitch. Interesting point.