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Arwen Evenstar remained also, and she said farewell to her brethren. None saw her last meeting with Elrond her father, for they went up into the hills and there spoke long together, and bitter was their parting that should endure beyond the ends of the world.
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This is from the chapter: Many Partings, probably the quote you're referring to.
Hardly anything is said of the two brothers anywhere, but I believe in the appendices it says that the sons went back to Rivendell and were presumably involved in the errands concerning Mirkwood, re-establishing Mirkwood as Eryn Lasgalen with their grandfather Celeborn and obviously with Thranduil up north. This was after Galadriel (again with Celeborn I believe) effectively destroyed Dol Guldur after the fall of Sauron and the evils of Mirkwood were largely dealt with.
After that, Celeborn lived in Lorien for a while and then went back to Rivendell presumably because there were still Elves living there (as well as his grandsons) and also many valuable artefacts and documents to guard, maintain and update. The last we hear of him is this quote from the prologue:
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"but there is no record of the day when at last he sought the Grey Havens, and with him went the last living memory of the Elder Days in Middle-earth."
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This does not say anything about Elladan and Elrohir, but seeing as they were obviously close family and also they had been living together in Rivendell for many years, estranged from Elrond, Celebrian and Galadriel (father/son-in-law, daughter/mother and wife/grandmother respectively) I would like to assume that the sons left with Celeborn. After all, if this is even after Arwen had died, there would be nothing left for them in Middle Earth and no real memory of their kin, lineage and importance. They would have probably become accustomed to living with their grandfather and I doubt they or he would have wanted to make the journey West alone (presumably with many treasures from Rivendell) so I personally think they went over to Valinor with Celeborn after they felt their 'love affair' with Middle Earth was over.
More similar to the Dunedain than Elf Lords in many ways, they roamed the lands often and no partners were ever mentioned concerning them. The Rangers will have disappeared once Aragorn became King, so really was there any good reason they would have chosen mortality?