It makes sense because Gollum is already pretty much two people. Why shouldn't I be the other one?
I like the way Gollum talks, incidentally, and I'm sure Tolkien did too. It occurs to me that even the fact that he has at least two personalities (Gollum, that is) doesn't fully account for his weird speech. He says, for instance, "We hates it forever," which is odd not so much because of the pronoun as because of the verb: one would say neither "I hates it forever" nor "we hates it forever," because "hates" can go only with the third-person singular -- only with "he, she, or it hates it forever." So Gollum's code-switching isn't just between singular and plural; sometimes (making the reasonable assumption that when it came to language Tolkien didn't do anything arbitrarily) he is talking to (or about) himself in the first person and the third person at the same time. This sounds exhausting. He also talks to himself in the second-person singular by addressing the absent Precious (or precious; if I had been Tolkien's copy editor I would have given him a stern talking-to about keeping his capitalization consistent) much as I do it by addressing Walter, say. All of which is to say that Gollum's constant soliloquizing is a means of both affirming and negating his existence; or, to put it yet another way, he hates himself so much that he tries to annihilate himself with grammar. The idea of a person so spiritually bewildered that he doesn't even know how to address himself is wonderful, so I take back that "big papist" remark of the other day, because maybe only a Catholic linguist could have given us the idea that souls need grammar too.
This being the case, wherever you are now, keep your capping consistent, OK, Professor?