The phrase "local diocese" refers to a geographical area that is overseen by a bishop, who is the leader of a specific region in the Catholic Church. Full definition
«We expect all Catholic schools to comply with the code and local dioceses provide support for schools to do so. (educationbusinessuk.net)
It is fair to say that we have reservations about schools being compelled to become academies, believing that the decision should be made by both the school and the relevant local diocese. (schoolsweek.co.uk)
Most of my reading online shows its local dioceses and to my arch diacoses in Mobile, Alabama not to the Vatican so much. (religion.blogs.cnn.com)