That's funny, I went out to the Arch on a moonless night when I lived there and I had it all to myself. It really depends on the time of year. Oddly, I've never encountered another photographer when I've been shooting in JT, but I've had nights where I've had to dodge them when I was leaving or going in. I remember one night when the MW rose about an hour before the full moon. I went, shot some star trails and a MW shot, and on my way out I was dodging tripods left and right, in my car, while driving down the road. Tripods were stacked in the street like runners staggered at a track meet. It was nuts. People were everywhere.
National Parks are pretty much like that from my experience. When we went to Yosemite people were setting up to shoot half dome hours before sunset. I took a picture of all the photographers at Glacier Point, it was insane. It was like the kind of crowds you see at parades or fireworks shows, only they all had tripods. If you're patient, and willing to sacrifice a large portion of your day getting to the "good spot" before anyone else then you can still get those shots. Personally, my wife and kids would kill me if I insisted on doing that. I'd rather be getting portraits of my kids at golden hour than standing shoulder to shoulder with 50-60 other photographers hoping to get the shot. Bear in mind too that some photographers aren't exactly considerate in those situations either. It's just not worth it to me.