Nothing is written about Elladan and Elrohir after the War of the Ring except that they remained in Rivendell, and Celeborn joined them there. Perhaps they chose mortality, or they had not yet decided. Perhaps they sailed to the West with Celeborn.
I imagine they aided Aragorn until he had subdued all of Gondor's enemies and saw Arwen as much as they could before finally sailing to Aman. Maybe it's just because I think it would really suck for Elrond to lose all his children.
What do you think they did? There are no right or wrong answers here.
I thought they had went with him, though who was left to govern Rivendell?
I would assume Rivendell was then either abandoned or given over to any Dunedain in the area after all the elves left.
I don't believe it is actually stated anywhere exactly their end choice whether to cross over or not,I know they stay in Middle-earth for a bit, but we don't know why for sure, and whether they actually chose mortal life or not is also unknown... Basically a lot of a grey area with them sadly! I'd like to think they chose to sail west in the end, but not until after Aragorn and Arwen passed away,
I'm inclined to agree they'd watch over Aragorn and Arwen for a while.