I think that no matter who's land it is...It was the State of NH that conducted the search and they're the ones billing the hiker.
This is a quote from the NH Forest Service website. There's a bunch of stuff on rescues there (of hikers, not paddlers), but this is all I could find about being charged for one - it's pretty vague. For more go to hikesafe.com
"In search and rescue scenarios, New Hampshire Fish and Game Department is the lead agency and is in charge of coordinating the rescue efforts. Typically, search and rescue crews are made up of New Hampshire Fish and Game Department Conservation Officers, White Mountain National Forest personnel and volunteers who venture into the backcountry in order to find lost and injured hikers.
Many times these individuals are putting themselves at personal risk in order to find someone who has perhaps used poor judgment and stumbled into a dangerous situation, especially when there is a severe weather situation.
Hikers should also know that if they need to be rescued as a result of their irresponsible actions, they can be charged for the cost of the event, which averages in the thousands per rescue. Annually, New Hampshire spends $260,000 on rescues"