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.)
It's too much of a risk of a conflict of interests - even when trying not to, as an insider you might give undue advantages to the student, or, even if not, other students might argue you did. You probably have qualified friends though who could help your student, and you could consider referring your student to one of them (they, too, need to of course check if this is ok under the terms of whatever contracts they are employed by by your university, if any).
Whatever you choose to do, it's a good idea to discuss it with the professor you TA for. Always be transparent, and avoid to be wrongly perceived as trying to hide your actions from your superiors.
I don't think it matters if you aren't grading, this is a big conflict of interest still. It is your job to help the students, so you shouldn't accept money to help some students more than others.
Make an effort to point him to a tutor who is not involved in the instruction of the course in any way. You don't want to look bad in the eyes of the other students, and you certainly don't want to jeopardize your TA-ship for a few bucks. You can always tutor students who aren't in your class if you need extra money.