Depends on which coast you live. West coast, central US lot's of government land (BLM, Forest Service, National Parks, etc.). The east coast, not so much. Most of it is private. It was a shock to me when I moved from Montana to Georgia.
The accessible woods are often private property, though, at least on the east coast. BLM and other state/federal lands are fantastic, and they're certainly plentiful out west. But lots of that land demands serious hiking or backpacking, and even the land that's easy to walk into isn't generally accessible without driving. It's not an option for inviting a friend over after school to explore the woods.
The land that is accessible by foot or bike is much more likely to be privately owned. And when it is public space, it's often a town or state park - which is quite likely to come with enough curfews, enforced bans on unaccompanied minors, and other child-unfriendliness that it's functionally private.