From Letter No. 151
* It is not made explicit how [Arwen] could arrange this. She could not of course just transfer her ticket on the boat like that! For any except those of Elvish race ‘sailing West’ was not permitted, and any exception required ‘authority’, and she was not in direct communication with the Valar, especially not since her choice to become ‘mortal’. What is meant is that it was Arwen who first thought of sending Frodo into the West, and put in a plea for him to Gandalf (direct or through Galadriel, or both), and she used her own renunciation of the right to go West as an argument. Her renunciation and suffering were related to and enmeshed with Frodo’s : both were parts of a plan for the regeneration of the state of Men. Her prayer might therefore be specially effective, and her plan have a certain equity of exchange. No doubt it was Gandalf who was the authority that accepted her plea. The Appendices show clearly that he was an emissary of the Valar, and virtually their plenipotentiary in accomplishing the plan against Sauron.
As for Elladan and Elrohir, they certainly remained for a certain time after Elrond sailed; Tolkien wrote (can't find the cite) that after Galadriel left, and Lothlorien started to fade, Celeborn went to Rivendell to live with them, for a while.
The impression I get is that here as with so many other matters around the edges Tolkien never really made up his mind, and even his 'definitive' statements may conceal complexity when expressed elliptically or briefly. Suffice it to say that Elrond's children got to make the Choice
at some point, but it's unclear when.