Ooh boy, this goes into some tricky areas where Martin is just not very consistent about Valyrian swords, because there are arguments that go either way: House Corbray lost Lady Forlorn to Robar II Royce and then supposedly got it back after the Battle of Seven Stars, which points one way, but Gwayne Corbray died on Redgrass and the sword somehow got back to the family (possibly because the loyalists retained the field and Daemon forbade looting of his honorable foe?).On the other hand, plenty of Valyrian swords were acquired by the ancient right of Acquiris Quodcumquae Rapis. The Harlaw blade Nightfall was won in battle, Red Rain was stolen by Hilmar “the Cunning” Drumm (with the aid of a blackjack), etc.

So I think this is something of a legal/cultural controversy: on the one hand, returning the arms and armor of a worthy opponent is a big part of the Knightly Code; on the other, Valyrian steel swords are literally beyond price and few people are going to have the moral strength to turn one down.