Sadly you don't actually have to be qualified or insured to carry out these treatments on the public, but it is obviously in yours and your client's best interest if you are just in case something went wrong and they decided to sue you.
Have you got your own student insurance? If so, check with them as they do vary a bit. When my daughter was at college we checked as she wanted to offer eyelash extensions. Because she had already completed that and got a certificate she was ok to charge the public.
I think for students you can offer certain treatments that you have received training for but you aren't allowed to charge full price until you have actually qualified and you need to make your clients aware that you are a student. But do check to make sure you don't have to have a tutor with you when you carry out the manicures.