I've been wheeling and dealing on here a lot the last two years for ADR, and have done it both ways.
Some people wanted me to call and change it to their name, others I just gave the ADR number and my last name to them, and they showed up in our place.
I've not had a problem so far either way, it's whichever you prefer. I do have a 'right of first refusal' policy, though. If someone gives me an ADR and later I decide I can't use it or want to swap it, I usually PM the Diser who gave it to me first and make sure they don't want it back or need it to barter with.