Since this thread has started to become an echo chamber I did visit the Town of Bethlehem today to get the property cards and inquire about if any of the parcels were in current use with the recreational "bonus" applied
As you can see by the attached blown up Tax map the answer is not as clear as I would like it but I agree with Tim that the current route appears to cross land owned by the owner of the lot at the end of the road. I have left that property owners name off this discussion as at least on this forum it has not been brought up and really isn't germane to the discussion. Note that the town of Carroll has a reservoir located on lot 10.2 (or possibly lot 10.3) but the town does not own the land.
View attachment 5952
As shown with my red line, the winter route crosses the bridge over to the SE corner of the land owned by Seven Dwarfs. It then most likely crosses over to the NE corner of Doug and Mary's lot but then it most likely transitions quickly to lot 10.3 which is owned by the owner of lot 10.2 who reportedly is posting no trespassing signs on lot 10.3. My memory of the winter route is it roughly parallels the river albeit drifting away from it as you head south and then regaining it as you approach the bridge. The best place to see this is using Strava. Using a scale that triangular wedge of lot 10.2 is 700 wide on its southern boundary and I believe the winter route is closer than 700 feet to the river (Strava indicates that pretty well). Thus if the property boundary between lot 9 and lot 10.2 were marked in the woods and someone walked on the west side of the line for 1254 feet south, they would never enter the disputed lot. Once past that point they have entered federal land and have the right to go anywhere they please. Using Google Earth imagery this route appears to steer clear of the developed portion of Doug and Mary's lot so it is possible route that respects lot 10.3 wishes.
I asked the town clerk if lot 10.3 is in current use and it is, but the owners did not take advantage of the slight reduction in taxes to leave the land open for recreational use (note that I am not familiar with the details on the property card that would show this distinction but the clerk indicated that the recreational "discount was not taken. She did indicate that Doug and Mary's lot did take advantage of the recreational discount on the current use portion.
I did not elect to go diving into the deeds as that is much more difficult effort. There is history of public use of this route by the public when the trailhead was located at the end of the road, thus prescriptive rights (adverse possession) is something that could come up. I expect that would be stretch as I was advised by a lawyer that generally the prescriptive rights have to be ongoing and if the reason for the use goes away, the possible adverse possession rights will extinguish. Generally a public agency or non profit would avoid trying to support exercise of these rights as it potentially can blow up into causing issues elsewhere.
So now the subject of posting of private land comes up. I have never seen a clear definition of how extensive the posting effort needs to be but NH F&G has this on their website
Under state law (RSA 635:4), the legal manner of posting calls for posting durable signs with any words describing the physical activity prohibited, such as "No Hunting or Trespassing," in letters at least 2 inches high, and with the owner's name and address. The signs may be no further than 100 yards apart on all sides of the property and shall also be posted at gates, bars and all commonly used entrances. To date I am not aware of any reports that the portion of lot 10.3 west of the river (the most likely point of entry) has been posted "no trespassing", it may have been in the past but this posting needs to be actively maintained. At a minimum I would expect there would be a no trespassing sign posted at both the northerly boundary line and the southerly boundary line with possibly several located along Doug and Mary's line to meet the statute. A private property sign does not meet the statute. I think it good idea on the owners behalf to put up a sign indicating its private property in hopes that individuals would show respect for it.
Without being able to talk directly to the landowner of 10.2 and 10.3, it appears as though his effort seems to be focused on the east side of the river which is his right on his property. The turnaround appears to be a public right of way but it not worth quibbling about it as there is viable alternative. I would speculate that ongoing abuse to his land probably occurs year round, its not unknown for folks visiting the area to just drive down a rural road in the evening and set up camp along a nice piece of river. Frequently they are bad guests and leave a trace such as fire rings and trash. As a landowner, that can get old quick especially when the tax bill arrives. I have no doubt that lot 10.3 gets the brunt of this and the convenient road to the reservoir probably makes it more likely the owner has issues. The hiking public also has potential for some blame as some individuals and groups are not good guests on occasion and some individuals can be quite arrogant about their perceived rights.
and VFTT could offer this as goodwill gesture. 
