There's where you can
WALK and where you can
CAMP. I'm talking about where you can
CAMP. The only countries in Western Europe where there's a lot of freedom to camp are Scotland, Norway, and Sweden.
In the Alps, where there is the most dramatic scenario, in France, Switzerland, Italy, Austria, there aren't a lot of areas where you can just pick a spot and camp there.
You don't need a permit to stay in a hut/refuge, but you usually need a reservation, which is sorta the same thing, a limit to the number of overnight visitors. Effectively a permit.
There are very very very few places in the USA where you need a permit to just hike for the day. Places where they are required are either very vulnerable like "The Wave" in Utah or dangerous when overcrowded like Angels Landing in Zion.