module_function makes the given instance methods private, then duplicates and puts them into the module’s metaclass as public methods. extend self adds all instance methods to the module’s singleton, leaving their visibilities unchanged.

module M
  extend self

  def a; end

  private
  def b; end
end

module N
  def c; end

  private
  def d; end

  module_function :c, :d
end

class O
  include M
  include N
end

M.a
M.b  # NoMethodError: private method `b' called for M:Module
N.c
N.d
O.new.a
O.new.b  # NoMethodError: private method `b' called for O
O.new.c  # NoMethodError: private method `c' called for O
O.new.d  # NoMethodError: private method `d' called for O