Some Key Points Missing
The current answers here have missed some key points that I believe help resolve a number of the questions regarding the plural pronoun uses.
"We" likely never references the Ring
Let me reproduce the same quote from the currently accepted answer that was used to make a case that "we" referred to the Ring from The Hobbit. I've bolded all the places where the Ring is being referenced and kept italics for the plural first person pronouns, then also highlighted the three times
precious is used by itself.
"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."
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,
"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!"
A number of points show that "we" in this passage is never referencing the Ring.
- Clearly, Gollum refers to the Ring multiple times in the third person as "it" (x7) and "the present" (x2) and "precious present" (x1). Some of these are in the immediate context of using a "we" or "our."
- In the final paragraph noting the Ring, Gollum even is postulating that a goblin will get a hold of the Ring, and then says "We shan't ever be safe again" and "Not even our clever eyeses will notice him," which are both clearly indicating the plurals are referring to the split Sméagol/Gollum persona.
- Both sides of the split persona use the "we," as seen in the first two paragraphs of the exchange and the last two, each being a different speaker, but each using "we."
precious term here, when used as a nominative, is the special term of one side of the personality (Gollum) towards the other (Sméagol), though that is not as clear here as in later writings where it is clear Gollum uses that term. However, the last paragraph "Then let's stop talking, precious" clearly shows the "precious" is referring to the other personality by the inclusive "let's" and the fact that the Ring is not in his possession and has not been part of the conversation. The one use of "precious" as an adjective refers to the "present" of the Ring.
I won't go so far to say that "we" is never a reference to the Ring in Gollum's speech, as I have not examined all the cases. But clearly here it is not, and I think likely in most all cases, if not all, it is not.
The initial Sméagol use of "us" may be the first manifestation of the split
In multiple comments and his answer, Wad Cheber makes it clear he believes Sméagol always spoke with the plural. The key evidence he posits is:
It has nothing to do with split personalities or the Ring's effects on him. It seems that this is just the way he always spoke. Smeagol didn't have a split personality until after he took the Ring as his own. Yet when Deagol first finds the Ring, what are the first words out of Smeagol's mouth?
"Give us that, Deagol, my love," said Smeagol, over his friend's shoulder. '"Why?" said Deagol. '"Because it's my birthday, my love, and I wants it"...
- The Fellowship of the Ring, Book I, Chapter 2: "The Shadow of the Past"
But this is by no means an open and shut case. The following points indicate this may well be the initial manifestation of the split:
- Only once does he use the plural to reference himself, using "my" (x3) and "I" (x1) also in the passage.
- That initial "us" comes immediately upon the Ring's first influence upon Sméagol; the Ring is already drawing him (and Deagol) to covet it and keep it. This strongly indicates that the Ring's influence may indeed have something to do with the manifestation of "us" in the request. The Ring influences from a (short) distance, so Sméagol not yet having had possession of the Ring does not mean its influence is out of scope for this encounter.
- Sméagol was clearly of "weak" character, being quickly and easily influenced by the Ring to kill Deagol over it.
However, based on that passage alone, either Wad Chebar's view or the view here could be correct. But there is one more key Tolkien gives to his thoughts on this...
"I" is specifically noted as being used of Sméagol when in his "good" form
In The Two Towers we find this during the exchange when Sméagol is trying to convince Frodo to not go to the Black Gate (bold added; italics original):
Frodo felt a strange certainty that in this matter Gollum was for once not so far from the truth as might be suspected; that he had somehow found a way out of Mordor, and at least believed that it was by his own cunning. For one thing, he noted that Gollum used I, and that seemed usually to be a sign, on its rare appearances, that some remnants of old truth and sincerity were for the moment on top. But even if Gollum could be trusted on this point, Frodo did not forget the wiles of the Enemy. The 'escape' may have been allowed or arranged, and well known in the Dark Tower. And in any case Gollum was plainly keeping a good deal back. ("The Black Gate is Closed")
This last quote distinctly indicates that the plural reference was related to Sméagol's alternate persona having influence over him, since Frodo observed (and Tolkien thus hints through him) that the singular "I" manifested in those places when the Gollum persona had lost influence.
"We" is primarily, if not possibly exclusively, a Sméagol/Gollum reference; it is a habit brought about from the first influence of the Ring upon Sméagol, but manifested more and more as he became isolated from society and had only "himselfs" as company.