Private community doesn't mean private access. From the Wiki: Clarification of the term “private”
"Note that access=private is intended to indicate that access is restricted, not whether the object is privately owned or not. Use ownership=private or operator:type=private to record this kind of status. For example, a privately owned road with public access may be tagged like any other road with public access – without access=* tag, or with the explicit access=permissive."