I've never used a dump trailer, but I do own a small dump truck, and I have gravel and top soil delivered to my place almost every year. Where to have it dumped is always something that I put a lot of thought into.
For topsoil, I've been having them dump it where I keep my round bales, and not an area that I consider nice grass. One day I hope to make it all gravel there, so I'm not too concerned what happens to the grass that gets covered by 8 yards of top soil.
For gravel, I have them dump on my driveway, where it's not in the way of our cars. My thinking is that what I don't pick up, will just add to what's already there. I would never want to dump gravel on any grass that I want to keep mowed.
I've never thought about dumping it into my bucket. I can't imagine how much longer that would take, or how much more effort would be involved. In my opinion, getting it done is my main priority, and not dealing with rock mixed into the grass is just going to create more work for me.
I've seen where contractors will lay down plywood, but I've never seen anybody use a tarp. Even with plywood, there is damage to the lawn that has to be dealt with. When it's dumped on the driveway, there isn't any clean up.