When you call a class member, Python automatically uses a reference to the object as the first parameter. The variable self actually means nothing, it's just a coding convention. You could call it gargaloo if you wanted. That said, the call to method_two would raise a TypeError, because Python is automatically trying to pass a parameter (the reference to its parent object) to a method that was defined as having no parameters.
To actually make it work, you could append this to your class definition:
method_two = staticmethod(method_two)
or you could use the @staticmethod function decorator.
The second one won't work because when you call it like that python internally tries to call it with the a_test instance as the first argument, but your method_two doesn't accept any arguments, so it wont work, you'll get a runtime error. If you want the equivalent of a static method you can use a class method. There's much less need for class methods in Python than static methods in languages like Java or C#. Most often the best solution is to use a method in the module, outside a class definition, those work more efficiently than class methods.