I wouldn't argue with anyone who stripped the "Plimpton Street" label for the trail - either the Sturbridge part or the whole thing!

In the absence of any other information, I look at what direction the 'private property' signs are pointed. If I am standing in the middle of a road/trail/path and looking straight ahead, and the signs are pointed directly at me and can read them without difficulty- I take that as a clue that the road/trail/path is private property. If I have to turn my head 90 degrees to read the signs, I then consider the signs indicating that the private property is off the side of the road/trail/path, and that the road/trail/path itself is public.