I think the question of why he left is closely related to why he was alowed to.
If you have the choice, I think you would want to (I would!)
And it seems like having owned the ring is enough; Sam, Frodo and Bilbo. as akind of reward for helping ME.
Frodo's injuries made him depart much earlier than Sam.
Interesting question: would Gollum be allowed to go west, personaly i think so, not clear about the motives.
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And with that he lifted high his sword and flames ran down the blade
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