the short answer
Personal trainer legal requirements usually come down to five things: a recognised certification, a registered business with correct tax handling, professional liability insurance, signed client contracts with a health questionnaire and waiver, and lawful handling of client data such as GDPR compliance. Personal training is rarely a licensed profession, but these basics are expected the moment a client pays you.
This article is general information for coaches, not legal, financial, or medical advice - certification, registration, insurance, and data rules vary by country and change over time, so verify the specifics for your situation with a local professional.