Hm, not quite sure of the legalities exactly, but we do all have to have insurance which covers us for professional indemnity as well as accidents whilst a client is on our premisis. If working in a salon or centre, the centre would be governed by normal health and safety law and would need to display the relevant poster from HSE, and those working at home/privately should display their current insurance certificate and their qualifications. When seeing a new client in their home for the first time, I always take my documentation to show them (but actually have never been asked for it!)
Further, we should abide by a moral obligation to observe safe practice in areas such as functionality of equipment, hygiene, proper conduct, ongoing learning etc. Having additional qualifications like First Aid is also sensible.
Hope this helps.