It also depends a lot on what the legal requirements are in your state. In a state with no mandate for testing for gifted kids (like where I live, MA), you generally won't be able to get public school testing at all for, "I think my kid is smart, let's find out more about it."
The other thing to really understand is that public school testing is designed to answer a legal question, "Is this kid eligible for whatever services we are legally mandated to provide if he's eligible?" Private testing, when it's done right (if I do say so myself) is much more about questions like, "What is this kid like as a learner? Where are his strengths and weaknesses? How does he approach difficult tasks? What could we do to best support his development?" and so on. Most of the folks who come to me (or people like me) for testing are less focused around the numerical scores and much more focused around the kid's approach -- how did he get to those scores?
Agreed also with other posters' concerns around the general non-kid-centeredness and not-always-GT-cluefulness of the school-based testing process. When I test, I clear a whole day, take the time to orient the kid, help him feel comfortable, take breaks / offer snacks as needed, etc.
