There are many reasons why people get piercings and tattoos. However some reasons are more healthy than others.
First, here are the more healthy reasons:
Simple enhancement. - Just adding some "spice."
Belonging to a particular subculture or community.
Tattoos show commitment. (Yes they can now be removed but it's not simple.)
Attracting a mate. (Girls like bad boys.)
These reasons are less healthy because they may indicate that you are not satisfied with your natural self:
Piercings and Tattoos show that you are tough and brave, because you went through pain to get them. - An emotionally secure person does not have to prove anything.
Rebellion - Acting out to protest feelings of discontent is not a healthy condition,
Getting noticed - Children should be made to feel special and unique right from the start.
Control - You cant choose your body, but you can choose what you do to it. - Being happy with your looks is essential, and there are more basic forms of enhancement which should be done first, (See: Do looks matter?")
"Are they decoration or mutilation?"
In American culture, some piercings are considered standard, some are considered edgy, and some are considered radical.
The person's age and the area of piercings comes in to play. There is a difference between a 12 year old girl getting earings, and a17 year old boy chaining his balls to his nose. Of course there aren't any real negative consequences of getting earings, but if you get a swastika tattooed on your forehead, things are not going to go well for you.
A radical or even edgy body modification can cost you jobs and turn people off, regardless of whether it's right or wrong.
The chances of regretting a tattoo later in life are higher if you get them while you are young. Kid's personalities naturally change as they develop.
"So how do I know?"
Luckily, the response is the same for all tattoos and piercings.
First, determine the reason. Some reasons are more healthy than others.
Second, make your kids aware of the negative consequences.
Third, weigh the reasons for getting tattoos/piercings against the negative consequences.
Summary
If your kids want to get a tattoo for one or more of the unhealthier reasons listed, then that issue should be addressed and resolved first before resorting to a tattoo/piercing. A body modification for a wrong reason will not fix the problem, it will only indicate that a problem exists, and can make things even worse.