No, at the moment mentors invited by their own students do not need to sign up or have a Publons profile in order to mentor in the 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 that you have mentored in the 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.
If you'd like to become a Publons community volunteer mentor for students without mentors then you do need to have a Publons profile and at least a few verified reviews on your profile. See this FAQ for how to become a Publons community mentor. If you want to invite your students to do the course with you then you also need to have a Publons account.