This segment is not private.  It is only private once you hit the military base further north of here.

Access tags on all points north of where it crosses onto base were set.
access=private
private=military

Because there is no access or signage until that point, this segment should not be marked private.  Let me know if you disagree.

https://www.alltrails.com/trail/us/washington/country-joe-s-to-nisqually-river-cliffs