A gun is made of plastic. It can not be safely sterilised as if it was it would simply melt or the components would break. Therefore it is only wiped down with an alcohol wipe or similar. This method does not safely kill off all of the pathogens that could potentially be harboured on the gun. Because of this you could be at risk of catching a blood borne pathogen from the people who were pierced before you with the same gun.
The ear studs unless you are willing to pay a hell of a lot more are usually made of 9k gold, gold plated or surgical steel. All of which by EU law a proper piercing studio is not allowed to use as they are classed as inferior metals loaded with things like nickel. Also, blunt force trauma!! Imagine piercing your ear with a blunt object.
That is what a gun does. An ear stud is blunt. It has to be to stay in your ear. It is forced through your ear and basically rips the layers on the way leaving a jagged shaped hole that is harder to heel. A needle is sharp, it cuts through properly leaving a neat circle ready for your implant grade jewellery to sit in.
We are educated on piercing, trained, experienced and it's what we live for. We do extra training, we throw away or sterilise what we use. We use high grade metals to keep you safe as standard. If you think your child needs a double gun piercing then your child may not be ready to get pierced at all. Pick safety over "it might hurt them less" it's your child's health and safety after all.