I know that my horse coughs a few times when I first get on him and start riding, but once he warms up, he stops. I call it "getting out the cobwebs." He never does it outside of that though, so I've never thought anything of it.
The current owner may be right, it might just be the hay, but a pre-purchase vet exam would be really smart. As far as paying, I would assume the person buying the horse would foot the bill just like if you are buying a car from some one and want to have a mechanic look at it first. Here in VA, it isn't too expensive for a check up, so I wouldn't sweat it. If it turns out to be something serious, $200 is a lot less than however much you pay for the initial purchase and the subsequent vet bills afterward.
Mommy to Minnie, Lilly, and Tanner.