For reptiles being euthanized at a veterinary clinic, in general the animal is first given an IM (intramuscular) injection of a sedative and then an overdose of a euthanizing solution directly to the heart or another organ in the chest/abdominal cavity. Due to the slow oxygen exchange rate (particularly while sedated), if the heart isn't directly injected it can take some time for the animal to pass. Anesthetic gasses are rarely used, as they take a long time to go into effect and reptiles have a habit of holding their breath when exposed to it.
Personally, I believe the momentary stress of a vet visit is well worth the peaceful death of your pet. I've had two chameleons euthanized over the last year and a half, and stand by my decision. Neither of my animals suffered, and their passing was swift and without any trauma beyond that first injection.
There aren't any truly painless or clean home deaths, and it's generally traumatizing for the person carrying it out. I grew up on a farm and have dispatched a large number of poultry and various mammals, and if you hesitate at all it gets really messy, really fast. Even with the experience I have, I'll never put down one of my own pets without the appropriate drugs unless it was an absolute emergency and moving the animal would cause undue suffering. Many vets also offer home euthanasia on request if you have a good relationship with them.
My two cents on the matter