Hunting is licensed by the states, and is usually relatively inexpensive for in-state residents. People from other states usually pay considerably more for a license. Big game or rarer animals may require special "tags" which come at larger fees and may be limited and awarded by lottery.
There are private game reserves. Georgia and other southern states have massive hunting preserves that can be a mark of old money, for instance. There are similar ranches in the west. And hunting clubs that anyone can just buy their way into.
And, to be honest, there's a lot of what is technically poaching- hunting on the private land of others without permission. That sort of thing isn't all that well policed, though state fish and game authorities are generally pretty tough on completely unlicensed hunting.
Here's a symptom of the cultural prevalence of hunting in the U.S.:
http://www.portal.state.pa.us/portal/server.pt/community/sta...
Most states will have such a listing.
There is also plenty of private land kept for hunting, which as you might mean to imply, is not cheap.
Interestingly enough, in Scotland although most wild areas are privately owned everyone has a right of access - you can wander about pretty much anywhere that isn't explicitly private or where you would cause harm.
Here in Scotland large parts of the country are privately owned sporting estates and shooting is not a cheap sport!