I agree with JJsMom that you should contact the school. The psychologist should have no problem talking with you and giving you some idea of how extensive the testing and reporting is. When talking with him I would definitely mention that you are wanting to do testing because you believe your child to be bright. At many schools if not most, the majority of the testing a school psychologist will do is for kids having difficulty, so it might be helpful to note that you are not looking for problems but rather the extent of his abilities, and that you would like specific information about his strengths and weaknesses in various areas. Feel free to ask more specific questions, too, about how the test works as far as continuing to ask questions when a child continues to get them correct (to make sure the psych wouln't stop when he saw your child was way above average, say). If you're not happy with the way the conversation goes or if you're not happy in the end with the results, then that is when I, personally, would pursue private testing.
We had our kids tested by the school, the first one because of possible LDs and the second two for early entrance and accelerations, and then also had one of our kids tested privately. Honestly, we got much better, more thorough data and reports from the school than from the private tester. But, obviously, that could be different for everyone. Now, our school district uses a test that isn't as widely used, so if we were interested in applying to DYS we would have to test again with a private tester, but since our goal was to make their public school education fit their needs, the free testing worked just as intended. HTH!