My Precious?

[Oddwen]

Here we get a little bit more into the "He hates and loves the Ring, as he hates and loves himself" bit.

I can almost see Gollum as the Ring come to flesh. He's devious, manipulative, and very cunning, while Smeagol is almost childish and naive.
Smeagol hates the Ring and Gollum because they almost "make" him do what he didn't/doesn't want to do. Maybe he's too childish to realize that it's his choice too, but giving in has become an age-old habit.
While Gollum hates Smeagol for holding him back.

And when a chance came it caught...Gollum, and it had devoured him. It could make no further use of him: he was too small and mean; and as long as it stayed with him he would never leave his deep pool again.

But then Gandalf says that Sauron sent forth his will, and the Ring "abandoned Gollum".
Is Gandalf referring to Gollum just because Frodo would know him by that name, or is he naming the evil half?

which side of Gollum's personality is making the promise to serve the master of the Precious?

I would say that both of them did it, Smeegs because he was frightened of Frodo and the Ring, and because Gollum could see a loophole in that promise.