Is it ethical to give (paid) private tutorials for a student in a module I am TA'ing?
Should I speak with the professor of the course myself about this?


I'd recommend telling the student you can't do it, and mentioning to the professor that you were asked but turned it down (to avoid any rumors).  There are massive issues here:


Being paid extra compromises your ability to grade the student's work.  It looks too much like a bribe, and the fact that the student will presumably stop paying you if the tutoring is ineffective (as measured by grades) creates an ongoing incentive to grade leniently or supply inappropriate degrees of help.  Even if you feel you can avoid bias, the apparent conflict of interest is so strong that other students will almost certainly be upset.  Think about it this way.  If the extra money doesn't matter to you and you have enough time, why not help the student for free?  The fact that money is changing hands at all indicates that the money matters to you (as it would for most people), and that is evidence of dangerous incentives.
It could look like you are extorting money from your students.  This is the flip side of the first point: in addition to your incentive to offer too much to those who pay extra, you now have an incentive to offer too little to those who don't hire you as a tutor.  Some students always grumble about how unhelpful their TA is (regardless of the facts), and you really don't want them speculating that your supposed unhelpfulness is intended to pressure them into paying for extra tutoring.
I'd bet this is a major violation of university rules.  Of course you'd have to look into your own case, but I'd be shocked if any university allowed course staff to accept paid tutoring from their students.


I'd be wary of asking the professor whether this is acceptable.  For most questions, asking can't hurt, but here the ethically questionable aspects are so strong that asking whether it is OK could look bad.  If you ask, you should be sure to make your understanding of the ethical issues clear, to avoid giving the impression that you think it's fine and just want to check whether the professor has any objections.

If the student needs a tutor, I'd recommend pointing him towards official university channels.  For example, perhaps your department maintains a list of students who would be willing to offer tutoring.  The important thing is to avoid any appearance of conflicts of interest.  Recommending hiring your friend instead of you is better than accepting the job yourself, but it could still look awkward.  (Students might wonder whether you were receiving a commission or kickback from your friend, or whether securing jobs for friends was enough of an incentive to influence your behavior.)