The recipient, no exceptions. If there's no gift receipt, sell it or give it away instead.
I think there ARE sometimes exceptions - if a gift is clothing that does not fit, and was purchased from a store not present in the recipients locale, then I think the giver needs to handle the return/exchange. This happened to me an awful lot when I was young, where I live in California, and the relatives were in New York, and bought lovely things in the wrong size from a local store that wasn't IN Los Angeles or San Francisco...
You bring up a good point.
I always try to determine where a recipient likes to shop and if there are those stores near them. I have friends on the East Coast and check with them to make sure if i send them something other than a gift card they can easily return it.
I do almost all of my gift buying online as I am not free to hit the malls due to my dad's health and having no one to stay with him while I'm gone.
In the case above where I buy something through Amazon, it would entail shipping it back to Amazon so I guess the onus would be on me in that case. If I can buy something online from a store I know they frequent then I'll either buy a gift or a gift card for them. In that case I think they should physically take the item to the store themselves. A lot of the retailers here allow you to bring things back to a store even if you bought it online and that does make it easier.
In the case of the OP I'm replying to in this thread, that's a tough one. I would surely rather ask if you had a store where you live where I was planning on purchasing and if you didn't I'd ask you to suggest a store or I'd buy a gift card for a store that is in proximity to you.