We usually choose BLM, DNR and NFS lands, federal or state lands where you can find a spot to boondock without paying fees, and where you get relative solitude. You can't just camp anywhere, though. Camping off of a road is frowned upon as is starting your own fire ring from what I understand.
When the government shutdown was in effect last October, we were in Great Basin National Park. The campgrounds were full so we camped in a parking lot. No one said a thing and the next day we were all kicked out. We also camped at Lunar Crater in Nevada by the dry lake; no one for miles. Later, we were exploring the Eastern Sierras and the USFS campground in McGee Canyon was shut down so we went up the road and camped in a turnaround. We also camped overnight in a closed USFS campground because the state park had closed down. Several others camped, too and no one came around to kick us out. Another spot was a parking lot where where fishers put in their boats somewhere near the California/Nevada border. We had planned to camp overnight at an abandoned gold mining area (up a winding rough 4x4 single lane road) but a couple of locals asked us if we saw some big black bears up there lately as they were hunting them.
It is very helpful to be self-contained and depending on your power needs/capacity, if you find a fine spot, you can stay for awhile. No reservations needed. Our dog can run and we can watch amazing sunsets/sunrises.
As far as determining a place to boondock, we look at greatly magnified Google Earth maps if we are preplanning to see pullouts on off roads or clearings. Just be sure that you are aware of your surroundings and don't camp in a dry wash or near rockfalls or other dangerous areas. Other times, it is serendipity that leads us to a place. When we go to a new area, we arrive early and scout around, as much for rockhounding and/or photography opportunities as a place to stay. Having a 4x4 high clearance vehicle does help.
We have never had to share our spot because we camp in remote areas. We have camped in areas where logging debris were piled, behind gravel piles on presumably road department storage areas, off a dead end road where it just ended (we were tired and it was dark). We awoke the next day and found out we were on BLM land that was leased by local ranchers. In the morning they drove by but no one said a thing.
Boondocking is an adventure. What we like about it, other than the quiet and remoteness, is being able to photograph right after stepping out the door of our TC or maybe hiking a bit further to get the right angle. No busloads of tourists asking you to take their photos. We have mostly quit going to national parks for that very reason. And, because we don't want to leave our dog in the truck for a photo op where he can't go. Our planet has some beautiful areas that belong to all of us that are less frequently visited. San Rafael Swell, Greater Canyonlands, eastern Oregon, most of Nevada (except for military bases), are yours to explore.
See ya out there.