We generally don't use the term "wild camping" here in the US. The official term is "dispersed camping," referring to camping outside established campgrounds, whether backpacking or traveling by vehicle.

On private land, permission of the landowner is required if you don't want to confront the local sheriff or furious landowner with rifle at 2 am. Most likely you won't get permission, because many people these days associate camping with homelessness rather than recreation. There are, of course, private campgrounds, although many tend to be limited to recreational vehicles. Unlike Europe, we regard camping as recreation, not budget travel, so you won't find campgrounds near towns or cities or accessible to public transportation.

We of course have a wide variety of public lands in the US, ranging from city parks to county parks to state parks and forests to national forests (or Bureau of Land Management areas) to national parks. Rules depend on the local land management agency. Many areas are day use only, and many others don't allow camping outside an established campground. Since individual national forests and national parks have widely different rules, even within a single forest or park, you really have to research the local jurisdiction. In most national parks, backpackers (what you call trekkers) need a backcountry permit which specifies exactly where and when they can camp. National forest wilderness areas (my preferred places) specify party size, how far you can camp from lakes, other water sources and the trail. There may be special restrictions or permits for more popular areas. These special permits are limited and generally need to be reserved months in advance.

Especially out here in the western US, there are usually campfire restrictions, since hiking season coincides with high fire danger season. In popular areas you generally can't build a fire within 1/4 mile of a mountain lake, or above a certain altitude, to avoid the proliferation of unsightly fire rings or the scarcity of fuel. In times of high fire danger, not only are campfires forbidden but also stoves burning alcohol or wood.

It is important to research the specific areas you will be visiting to find out the rules for the specific locations you'll be!