Not really private islands, and they don't really retire either.
When a star is old enough and has lost enough power, it can go to one of the islands on the Eastern Ocean. While there, it recovers its strength and becomes young again by eating "fire-berries".
"When I set for the last time, decrepit and old beyond all that you can reckon, I was carried to this island. I am not so old now as I was then. Every morning a bird brings me a fire-berry from the valleys in the Sun, and each fire-berry takes away a little of my age. And when I have become as young as the child that was born yesterday, then I shall take my rising again (for we are at earth's eastern rim) and once more tread the great dance."
Voyage of the Dawn Treader, Chapter XIV: The Beginning of the End of the World
So when he's recovered, he'll go back to being in the sky. It's kind of like a sabbatical for stars.
Also, stars can come down for other reasons and still be fully able to fly in the sky.
"Even in your world, my son, that is not what a star is but only what it is made of. And in this world you have already met a star: for I think you have been with Coriakin."
"Is he a retired star, too?" said Lucy.
"Well, not quite the same," said Ramandu. "It was not quite as a rest that he was set to govern the Duffers. You might call it a punishment. He might have shone for thousands of years more in the southern winter sky if all had gone well."
So it's not really a retirement, it's more of a temporary break. And the last quote implies that stars prefer to be in the sky versus on the ground, because Coriakin was punished with coming down.
For those who don't remember, the duffers were the (for a time) invisible and one-legged creatures. Coriakin was supposed to be guiding them.
Finally, it's also unknown what punishment merits a star coming down.
"What did he do, Sir?" asked Caspian.
"My son," said Ramandu, "it is not for you, a son of Adam, to know what faults a star can commit. But come, we waste time in such talk. Are you yet resolved? Will you sail farther east and come again, leaving one to return no more, and so break the enchantment? Or will you sail westward?"