Ndtyhij

In The Return of the King, when the Fellowship finally parts ways, Pippin says that he wishes they each had palantiri so they could communicate with each other over long distances. Aragorn responds:

Only one now remains that you could use, for you would not wish to see what the Stone of Minas Tirith would show you. But the Stone of Orthanc the King will keep, to see what is passing in his realm, and what his servants are doing.

-The Lord of the Rings, The Return of the King, Book VI, Chapter 6: "Many Partings"

I assumed that the King in question was Eomer, the King of Rohan. But in a comment on another question, someone suggested that Aragorn was talking about himself in the third person (i.e., "But the Stone of Orthanc I will keep, to see what is passing in my realm, and what my servants are doing"). This seems strange to me, and we never really hear Aragorn speak this way elsewhere in the story.

It also isn't clear why Aragorn would want two palantiri - one should be enough.

Is there any evidence to clarify this issue? Did the Stone of Orthanc go to Eomer or Aragorn?

Aragorn keeps possession of the stone. The following quote from Many Partings given by @MattGutting is enough to establish this.

'Only one now remains that you could use,' answered Aragorn, 'for you
would not wish to see what the Stone of Minas Tirith would show you.
But the Palantír of Orthanc the King will keep, to see what is passing
in his realm, and what his servants are doing. For do not forget,
Peregrin Took, that you are a knight of Gondor, and I do not release
you from your service. You are going now on leave, but I may recall
you. And remember, dear friends of the Shire, that my realm lies also
in the North, and I shall come there one day.'

To conclude from this that Aragorn wanted Eomer to watch over a Gondorian knight living in the jurisdiction of Arnor would be bizarre, and can be discounted. But, since Aragorn would clearly have the strength to control the Anor (Minas Tirith) stone, why did he need two? There are two further references that have a bearing on this. At the end of The Disaster of the Gladden Fields (Unfinished Tales), it is stated that Aragorn and Gimli searched Orthanc,

where he [Aragorn] proposed to set up again the palantir recovered
from Saruman.

Why here? Why not entrust the Orthanc stone to his closest and most powerful ally, Eomer,
and place it at Edoras? Why not place it in Annuminas, to stay in contact with his Northern realm, or even place it in Dale, since future threats to Gondor were likely to come from the East? This took some working out, but the answer is given in note 18 to The Palantiri (Unfinished Tales again).

The greater palantiri could look much further than the lesser; for the
lesser the 'proper distance' was of the order of five hundred miles,
as between the Orthanc-stone and that of Anor.

(Emphasis mine.) Only by returning the Orthanc-stone to its original place and assigning a suitable warden there could Aragorn establish a means of immediately finding out what was happening in Arnor. The major stones (Amon Sul and Osgiliath) were lost. A single minor stone held in Minas Tirith would have insufficient range.

There's no direct evidence. Orthanc, Isengard, and the Stone were of course the property of Gondor and therefore of the King. In that very chapter you cite, Aragorn makes a big deal of specifically ceding Orthanc and Isengard to the Ents:

[Gandalf:] 'But the Tower of Orthanc now goes back to the King, to whom it belongs. Though maybe he will not need it.'

'That will be seen later,' said Aragorn. 'But I will give to Ents all this valley to do with as they will, so long as they keep a watch upon Orthanc and see that none enter it without my leave.'

He says nothing specific about giving the palantír to anyone—in particular, he certainly seems not to have given it to the Ents (who probably wouldn't have wanted it in any case); and it seems unlikely that he would have just left it in a locked building. The context of the quote you give in your question, however, may give a clue:

'Only one now remains that you could use,' answered Aragorn, 'for you would not wish to see what the Stone of Minas Tirith would show you. But the Palantír of Orthanc the King will keep, to see what is passing in his realm, and what his servants are doing. For do not forget, Peregrin Took, that you are a knight of Gondor, and I do not release you from your service. You are going now on leave, but I may recall you. And remember, dear friends of the Shire, that my realm lies also in the North, and I shall come there one day.'

(Return of the King, Book VI, Chapter 6, "Many Partings"; emphasis added)

The fact that Aragorn first speaks of a king watching what his servants are doing, and immediately follows that by a reminder to Pippin that he is a servant of (that is, one in service to) Aragorn's kingdom, and that Aragorn may have cause to know what Pippin is doing and summon him, strongly suggests to me that Aragorn is speaking of himself here.

This is unclear.

Unfinished Tales implies that Aragorn did claim it for himself (bold is my emphasis, italic is in-text):

It must however be noted with regard to the narrative of The Lord of the Rings that over and above such deputed authority, even hereditary, any "heir of Elendil" (that is, a recognized descendant occupying a throne or lordship in the Númenórean realms by virtue of this descent) had the right to use any of the palantíri. Aragorn thus claimed the right to take the Orthanc-stone into his possession, since it was now, for the time being, without owner or warden; and also because he was de jure the rightful King of both Gondor and Arnor, and could, if he willed, for just cause withdraw all previous grants to himself.

^{_{Unfinished Tales Part 4 Chapter III: The Palantíri}}

However, it's not clear whether this refers to Aragorn claiming the Orthanc-stone in The Two Towers, rather than the line under discussion:

'There is one who may claim it by right. For this assuredly is the palantír of Orthanc from the treasury of Elendil, set here by the Kings of Gondor. Now my hour draws near. I will take it.'

Gandalf looked at Aragorn, and then, to the surprise of the others, he lifted the covered Stone, and bowed as he presented it.

'Receive it, lord!' he said: 'in earnest of other things that shall be given back. But if I may counsel you in the use of your own, do not use it - yet! Be wary!'

^{_{The Two Towers Book III Chapter XI: "The Palantír"}}

Unfortunately, it doesn't appear as though Tolkien ever resolved the ambiguity (or, indeed, if he ever considered it ambiguous); there's nothing I can find in Letters or History of Middle-earth that comes down one way or the other.

Aragorn, as the King, took possession of the stone. When he uses third person, I think he refers to himself but also to each King that will come to rule in his stead. The stones are the property of the King (which ever one is currently ruling).
As for the stone in Minas Tirith, I'm sure I've read somewhere that it became unusable after the incident with Denethor. It would only show some horrible image of a burning person, or some such. Gonna have a look at the exact quote.

It is unquestionably Aragorn who retained the Palantir, not only for the reasons outlined already, but also because Gondor retained the tower as an exclave

the Stewards retained under their own rule the Tower of Orthanc and the Ring of Isengard (Angrenost); the keys of Orthanc were taken to Minas Tirith, the Tower was shut, and the Ring of Isengard remained manned only by a hereditary Gondorian chieftan and his small people.

-- Appendix II, The Battle of the Fords of Isen, Unfinished Tales

The King over Isengard, and the owner of the Palantir - whose forebears placed it in the tower millennia beforehand, were one and the same: Aragorn II.