In addition to the other answers dealing with mortality in Valinor, it's clear from the books that the effects of the Ring are only temporary, and that they will wear off in time. For example, Gandalf's comment in Shadow of the Past:
he possessed the ring for many years, and used it, so it might take a long while for the influence to wear off...
So it's clear that:
- the effects of the Ring do wear off after you no longer have it,
- this wearing off takes time,
- how much time it takes depends on how long you have the Ring and how much you use it.
Looking now at the Ring-bearers, we see that
- Sam had it the least amount of time (excepting Déagol, who was killed almost immediately), 2 days maximum, and gave it up willingly, so it shouldn't be expected that it had much lasting effect on him.
- Frodo had it about 18 years, but the Ring completely claimed him at the end and was taken off him by force; that's obviously going to have a lasting effect.
- Bilbo had it about 60 years, only used the Ring for tricks (and escaping from the SBs), and also gave it up willingly; he was without it for almost 20 years when he went West.
- Gollum had it for almost 500 years, obtained it by force, used it for evil, lost it unwillingly, and was without it for close on 80 years at his death.
Obviously, in the light of Gandalf's statement, it's going to take longer for the effects of the Ring to wear off both Bilbo and Gollum. Even when we meet Bilbo in Many Partings they're still wearing off - he asks to see the Ring. Gollum of course still wants it right to the very end, and as such is still under its influence.
