I don't think the public school, itself does the testing, it's just under its auspices and will influence how you are treated in school. Independent will be outside of school and won't necessarily impact that or anything else as school might insist on its own testing for its purposes and the SAT, etc. will require more paperwork that your parents will have to complete, if you do not have the school doing the request for SAT consideration.

I would think about what you want the results for. Just someone saying you officially have "short term memory/recall and retention of information" difficulties won't do anything for you in your life; whoever you want some consideration of for assistance will have their own rules and hoops for you to jump through so I'd figure out who you might need, what school, college testing program, college/university you were going for and read what they have to offer and what it takes to get consideration, etc. and then decide. If it is primarily for learning though and in learning situations, I'd go for school testing.