When I bought my horse, the owner did all the riding, trot up etc and I stood with the vet told he told me what he observed at each stage of the process
When I've sold I've always done the riding the horses I've bought have always been through my friend who a dealer and I've I had them on trial first and then had a 2 stage vetting done.
If I was to buy privately off somebody I didn't know I would be present for betting and want to ride it myself too. I don't think it really matters to be honest though.
It's usually the seller, though the only time I've had a horse vetted (the recent addition!) the vet gave me the choice of whether I wanted to ride round or if I wanted the dealer to do so. I chose the dealer as I'd ridden her round before (obviously!) and felt we were suited, so I stood with the vet while he told me what he saw & thought of her movement etc. If I hadn't been sure of whether I was suited to the horse and wanted another opinion, I would have ridden her round and asked him what he thought, as he knows the horses we have now & what I'm "used to" etc - if that makes sense!