No, at the moment mentors do not need to sign up or have a Publons profile in order to mentor their student in Publons Academy. However, if you do sign up (it's free) you can also start building your public profile of verified peer review and editorial contributions, and your profile will also show how many and who you have mentored in Publons Academy. You can export a verified report of all your service contributions to your field and add it to funding and job applications. If you end up signing up for Publons after you've already mentored a student, if you verify the email address you used in the Publons Academy, your Academy activity will show up on your profile.
Students do need to register to have a Publons account to take the course. During the course they will start to complete their Publons profile and get the Academy post-publication reviews they write during the course added to their profile and dashboard depending on their chosen review privacy settings.