While it couldnt be clear what they are doing exactly there, except enjoying the presence of the Valar and the Maiar among the elves. I think that it could be answered what they are doing in a more general way.
Both <PERSON> and <PERSON> were the Ring-bearers, so it seems that they went there as a reward and, at the same time, because it was the only site where they could live without suffering from the influence of the ring in their lives (Think about <PERSON>)
From the Lord of the Rings, book 6, chapter 9:
It is mentioned a few times in the text:
... for the Third Age was over, and the Days of the Rings were passed, and an end was come of the story and song of those times...
And not only <PERSON> and <PERSON>, but even <PERSON> is invited to go, but not now:
Then <PERSON> woke up and opened his eyes. ‘Hullo, <PERSON>!’ he said. ‘Well, I have passed the Old Took today! So that’s settled. And now I think I am quite ready to go on another journey. Are you coming?’
‘Yes, I am coming,’ said <PERSON>. ‘The Ring-bearers should go together.’
‘Where are you going, Master?’ cried <PERSON>, though at last he understood what was happening.
‘To the Havens, <PERSON>,’ said <PERSON>.
‘And I can’t come.’
‘No, <PERSON>. Not yet anyway, not further than the Havens. Though you too were a Ring-bearer, if only for a little while. Your time may come. Do not be too sad, <PERSON>. You cannot be always torn in two. You will have to be one and whole, for many years. You have so much to enjoy and to be, and to do.’
I think that there was a text where <PERSON> goes to Valinor. I think that <PERSON> went too, because he is an elf and he managed to carry <PERSON> with him.
Other part of the reward would be, help them to clean the corruption from the One Ring.
The most clear proof of the corruption of the One Ring is from Book 1, Chapter 1 with the confrontation between <PERSON> and <PERSON>:
'Everything?' said <PERSON>. 'The ring as well? You agreed to that, you remember.'
'Well, er, yes, I suppose so,' stammered <PERSON>.
'Where is it?'
'In an envelope, if you must know,' said <PERSON> impatiently. 'There on the mantelpiece. Well, no! Here it is in my pocket!' He hesitated. 'Isn't that odd now?' he said softly to himself. 381
'Yet after all, why not? Why shouldn't it stay there?'
...
<PERSON> flushed, and there was an angry light in his eyes. His kindly face grew hard. 'Why not?' he cried. 'And what business is it of yours, anyway, to know what I do with my own things? It is my own. I found it. It came to me.'
'Yes, yes,' said <PERSON>. 'But there is no need to get angry.'
'If I am it is your fault,' said <PERSON>. 'It is mine, I tell you. My own. My precious. Yes, my precious.'
The wizard's face remained grave and attentive, and only a flicker in his deep eyes showed that he was startled and indeed alarmed. 'It has been called that before,' he said, 'but not by you.'
...
'Now, now, my dear hobbit! ' said <PERSON>. 'All your long life we have been friends, and you owe me something. Come! Do as you promised: give it up! '
'Well, if you want my ring yourself, say so!' cried <PERSON>. 'But you won't get it. I won't give my precious away, I tell you.' His hand strayed to the hilt of his small sword.
...
'I don't know what has come over you, <PERSON>,' he said. 'You have never been like this before. What is it all about? It is mine isn't it? I found it, and <PERSON> would have killed me, if I hadn't kept it. I'm not a thief, whatever he said.'
...
<PERSON> drew his hand over his eyes. 'I am sorry,' he said. 381