Cicero did have a slave called Tyro who served as his secretary and confidante, that's the closest example that I can give to a political staff but my inclination is that they did not exist in the modern sense. On the speech writers point it was considered very bad face for a Roman not to write his own speeches. Nero was lambasted for it because it was clear that he had not written his own speeches. The custom was different in Greece where orators like Lysias making some extra income as paid speech writers, for instance "On the Murder of Eratosthenes".

I'm not sure if this strictly fits what you're asking, but there is mention of nomenclators in the late republican times. These were slaves who accompanies candidates for office. Their job was to remember lots of names and whisper them to the candidate so he if they were personally known to him.

Nay more, why is it that you have a nomenclator with you? for in so doing, you are practicing a trick and a deceit. For if it be an honourable thing for your fellow-citizens to be addressed by name by you, it is a shameful thing for them to be better known to your servant than to yourself.

from M. Tullius Cicero. The Orations of Marcus Tullius Cicero

Seems the practice was looked down upon to some extent,