From the book The Two Towers, here's the Gollum/Sméagol dialogue. It's at the end of Book IV, Chapter 2, The Passage of the Marshes. I've modified it slightly to show just the speech, along with the name of the personality speaking, and I've emphasized the personal pronouns:
Sméagol: I don't know. I can't help it. Master's got it. Sméagol promised to help the master.
Gollum: Yes, yes, to help the master: the master of the Precious. But if we was master, then we could help ourselfs, yes, and still keep promises.
Sméagol: But Sméagol said he would be very very good. Nice hobbit! He took cruel rope off Sméagol's leg. He speaks nicely to me.
Gollum: Very very good, eh, my precious? Let's be good, good as fish, sweet one, but to ourselfs. Not hurt the nice hobbit, of course, no, no.
Sméagol: But the Precious holds the promise.
Gollum: Then take it, and let's hold it ourselfs! Then we shall be master, gollum. Make the other hobbit, the nasty suspicious hobbit, make him crawl, yes, gollum!
As you can see, "we" is used by the Gollum persona, and the Gollum persona is the one that constantly uses the "my precious" expression. The Sméagol persona uses I instead of we, and pretty decent grammar by comparison, and never uses the "my precious" affectation.
From the above dialogue, at first glance, it seems an open-and-shut case that the "we" refers to Gollum and Sméagol. But I don't think this is true.
The Gollum persona came about due to the influence of the ring, and constantly addresses the ring as "my precious", using "we" in conjunction with that. The above dialogue is a clear exception to the sort of thing we see in The Hobbit. Here's the longest extended monologue by Gollum from that book:
"It's no good going back there to search, no. We doesn't remember all the places we've visited. and it's no use. The Baggins has got it in its pocketses; the nassty noser has found it, we says."
"We guesses, precious, only guesses. We can't know till we find the nassty creature and squeezes it. But it doesn't know what the present can do, does it? It'll just keep it in its pocketses. It doesn't know, and it can't go far. It's lost itself, the nassty nosey thing. It doesn't know the way out. It said so."
"It said so, yes; but it's tricksy. It doesn't say what it means. It won't say what it's got in its pocketses. It knows. It knows a way in, it must know a way out, yes. It's off to the back-door. to the back-door, that's it."
"The goblins will catch it then. It can't get out that way, precious."
"Ssss, sss, gollum! Goblinses! Yes, but if it's got the present, our precious present, then goblinses will get it, gollum! They'll find it, they'll find out what it does. We shan't ever be safe again, never, gollum! One of the goblinses will put it on, and then no one will see him. He'll be there but not seen. Not even our clever eyeses will notice him; and he'll come creepsy and tricksy and catch us, gollum, gollum!"
"Then let's stop talking, precious, and make haste. If the Baggins has gone that way, we must go quick and see. Go! Not far now. Make haste!"
Notice that though we can posit two speakers, the grammar doesn't change. The term "precious" is used repeatedly by only one speaker, but the pronoun "we" is used by both. Other than the term "precious", there's no clear distinction between the two speakers as there is in the Gollum/Sméagol dialogue.
The Sméagol personality doesn't appear to have manifested itself for a long time, dozens or perhaps hundreds of years, until Frodo addressed Gollum by the name of Sméagol and briefly revived Gollum/Sméagol's memory of his old self.
Thus "we" refers to Gollum and the One Ring, and not to Gollum and Sméagol. The dialogue from The Two Towers is an exception to the rule.