Yes there is an EBMUD permit to use the trails but it's so loosely enforced that effectively access=yes because anyone from the public who wishes to hike these trails can.  Access=permit suggests that there's a restriction but in this case anyone who wants the permit can get one (there's no waitlist or limit) and nobody is guarding the trailheads to check for permits.  I've hiked every EBMUD trail and have never been asked to show a permit.  And access=permit is not consistently applied to all of the EBMUD trails (some show access=yes).  This creates problems for applications like wandrer.earth that are working with global norms where permit is considered to be restrictive and treated as private so the paths are not visible or considered.