I interviewed a shelter guy last week for a story about lost pets and we talked about ID methods. He says that no one recommends tattooing anymore, microchipping is the way to go.
With microchips, the companies do maintain a central database to call. With tattoos, I'm not sure there is a database anymore.
I think it's unlikely a puppy would be tattooed, then. An older dog might have a old tattoo.
If you really want to go with this for some reason, I have a vague recollection of suggestions that people use their social security number as the tattoo. What you'd have to think of was what made sense to someone who was stubbornly sticking to an outdated technology. They wouldn't use their cell phone number, I'm guessing - that seems inconsistent.
But it would make a lot more sense to just go with a microchip if what you need is a method to identify a lost dog with no tags.