The gift of men is to rejoin Iluvatar before the end of Arda.  Although we know from Beren's story that they may spend some time in the Halls of Mandos (purgatory?) first.  And we know from the fate of the Nazgul, Ar-Pharazon and his men, and the spirits from the Paths of the Dead, that even the trip to the Halls of Mandos can be delayed.  Although my theory about Ar-Pharazon and his men is that instead of them coming to the Halls, the Halls came to them.

What exactly happens to human spirits when they leave Arda no one knows, and Tolkien does not even hint at an answer.  But he was a devout Roman Catholic, so it probably was consistent with pre-Vatican II Roman Catholic beliefs.

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"‘I think he was a silly little man,' said Councillor Tompkins.  ‘Worthless, in fact; no use to Society at all.'

"‘Oh, I don't know,' said Atkins, who was nobody of importance, just a schoolmaster.  ‘I am not so sure: it depends on what you mean by use .'

"‘No practical or economic use,' said Tompkins.  . . .

. . .

"‘It is proving very useful indeed,' said the Second Voice.  ‘As a holiday, and a refreshment.  It is splendid for convalescence; and not only for that, for many it is the best introduction to the Mountains.  It works wonders in some cases.  I am sending more and more there.  They seldom have to come back.'"