It all depends on the service of your private lesson. If you are strictly giving lessons on Muay Thai, then you’re not any different than any other martial artist giving out private lessons. As far as anyone is concerned, there’s little in terms of acquiring certifications to teach martial arts, let alone offering private instructions.
If any liability issues arise, the general inquiries (as with all trainer of any kind) will look into your own training history (who you trained under, how long you trained for, was there anyone who gave you the okay to instruct, etc.). Then it will look into the nature of the incident (i.e. were you doing anything that would be considered out of your scope). If you were strictly doing Muay Thai with your client, then the questions will start to focus on the specifics of your client (e.g. were they physically ready for what you required them to do, were their medical history taken into consideration, etc.). Then it may fall to question the nature of the instruction (e.g. is this something that other instructors would have done, etc.).
If you already teach at the school, then I’d suggest talking with the school owner/operator (assuming you aren’t the owner/operator of the school) and see what the policies are with private lessons. You may already be covered for this.
I do know some places requires one to have a certification to teach combat sport classes, and some organizations gives these types of certifications; but keep in mind that it’s on select few combat sport (kickboxing being the most foremost of them). Even so, this type of certification is for teaching in a group setting. The scope for this type of certification is very limited.
There is no certification to my knowledge for martial arts. That’s the great thing about the arts its what you make it and choose to pass down to the next generation. However I would highly recommend getting some sort of national certification. This holds weight its fair to you and your clients health and safety. Teaches you a lot about exercise and the human body and different thoughts on fitness.
I was in the same boat as you in that I was teaching before becoming certified becoming certified was my best move and I would highly recommend putting in the time.
Hopes This Helps
It is not required to have a certification in anything to give private lesssons. However as it was said before being an athlete does not make a trainer be. It is important to have serious knowledge of medical, mechanical, physiology and teaching experience behind you in a certification.
That be said anyone can teach anything, however to be legally bonded in anyform of insurance policy you will need to be accrediably certified or work under someone who is to protect you and your client.
Hope it helps.