A would never recommend a school doing testing... A school isn't going to guarantee that the place will be quiet and that the test can be done with out some phone ringing or someone dropping in, any of those things can easily happen and easily throw a kids' concentration.
That's an odd generalization.
Second problem is suppose your child has a bad day, isn't feeling well or whatever... if that happens the day they are taking the test you just put a low score in the child's permanent record and with no child left behind policies their is no guarantee that it wont be available to every teacher the child has until they graduate. It will also be a problem if you later test him by a private place and he tests gifted... now the school has a gifted score and a non-gifted score... which will they use? Some might use the higher score, others that already have as many kids in the gifted program that they want will see the low score as a good way to keep him/her out.
My district's only limitation is that you can't re-screen a kid for giftedness for a full calendar year from the date of the last test. If the child re-tests later and scores much higher, that's the test they'll consider valid.
If you do the private testing you control the test results not the government funded school.
And here you've accidentally described one of the benefits of school testing... because the school will tend to view private results, especially where they're not familiar with the tester, with a bit of suspicion, since private testers do have a financial incentive to provide the results that the parents expect. When the testing is completed by the school, they trust their results.