When I hiked this trail a few months ago, it wasn't clear to me whether the signs were trying to say that the trail itself was private, or just that the land on the side of the trail was. That is, I don't know if the signs were saying to stay on the trail and not explore elsewhere, or saying to not go on the trail at all. (Though given that the trail ends at the pipeline, there isn't really much reason for anyone to take this trail.) This is just FYI; I have no objection to the edit.
Do you have more information on whether it's the trail itself that's private, or whether it really should still be called "Plimpton Street"? I'm honestly not even sure it should still have that name through the WMA at all, let alone this portion once one goes through it all, though I'm guessing (just a guess) that it was all called that at one point.