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Not American but Swedish, so:
In Sweden we have something called "Allemansrätten" (The right of all men/All men's right, essentially) which allows us to camp, hunt, climb, create fires, ride horses, pick berries and mushrooms, et c on both public AND private lands. The activities on private lands are allowed as long as we don't intrude on the land owner's privacy or financial interests in the area.
We are, in other words, allowed to put up a tent for a couple of days on someone's land as long as we don't disturb them or destroy their property. A distance of a couple of hundred yards from their house could be enough.
When I read the land it first seemed to me like this "public land" or federal land is to be treated in the same way, that it offers more freedom to the average person - but it's that not the case? Does it seem to depend on what kind of federal agency that owns the area?
I'm visiting your country soon and will probably do some hiking, so I'm wondering a bit about this.