This goal of wanting "the difference to be clear when the function is called" doesn't rub so well with the design philosophy of the language. Google "duck typing" for more information about that. The accepted inputs should be made clear by the docstring of the function, and that's all you need to do.
In python, when you want your input it to be a string or a dict, then you just write code which assumes the input will be an object which behaves in behave in a string-like way or a dict-like way. If the input doesn't, then you can try to handle that gracefully if you want, but usually you can simply leave your code to simply pop with an unhandled exception. This puts the ball back in the caller's court to either decide how to handle the situation, or realise that they were sending in bad data.
Type checks should be avoided in general, if really necessary it should be done with isinstance
rather than an equality check on the type like you have done. This has the advantage of being more flexible for inheritance situations.
def aFuncion(input_):
if isinstance(input_, str):
print 'you gave a string-like input'
elif isinstance(input_, dict):
print 'you gave a dict-like input'
aFunction('test')
In python3 you now have another option of using type hinting function annotations. Read PEP 484 for more details about that feature.