private_override.rb

class Foo
  def calls_bar
    bar + " from foo"
  end
  private
  def bar
    raise "define #{__method__} in subclasses"
  end
end
class SubFooPrivateOverride < Foo
  def initialize
    puts "the method scope of bar is still private in this subclass: call without self #{bar}"
    # The next call will fail
    puts "the method scope of bar is still private in this subclass: call with self #{self.bar}"
  rescue NoMethodError => e
    puts "#{e.class}: #{e.message}"
  end
  private
  def bar
    "private subfoo bar"
  end
end
class SubFooProtectedOverride < Foo
  def initialize
    puts "the method scope of bar is changed to protected in this subclass: call without self #{bar}"
    puts "the method scope of bar is changed to protected in this subclass: call with self #{self.bar}"
  rescue NoMethodError => e
    puts "#{e.class}: #{e.message}"
  end
  protected
  def bar
    "protected subfoo bar"
  end
end
puts SubFooPrivateOverride.new.calls_bar
puts SubFooProtectedOverride.new.calls_bar

Running this gives:

∴ ruby foo.rb
the method scope of bar is still private in this subclass: call without self private subfoo bar
NoMethodError: private method `bar' called for #<SubFooPrivateOverride:0x007ff725053100>
private subfoo bar from foo
the method scope of bar is changed to protected in this subclass: call without self protected subfoo bar
the method scope of bar is changed to protected in this subclass: call with self protected subfoo bar
protected subfoo bar from foo