When you first turn onto the Bedford Motorway it is signed as "private" but not restricted. After passing in a tunnel under the development's main road, there are signs that say "authorized vehicles only". You pass a large gated house and then there is a "private property, no trespassing" sign. At 2 miles is a sign specifying typical rules for public use, such as licensed vehicles only, which seems to indicate that the public is expected. The Cleveland National Forest web site lists the Bedford Road as open. I saw several other vehicles, motorcycles, and bicycles on the road and I went with the interpretation that the public was "authorized" and the "private property" refered to the land along the road.
"After the first glass, you see things as you wish they were. After the second, you see things as they are not. Finally, you see things as they really are, which is the most horrible thing in the world."
--Oscar Wilde on Absinthe