Why do you have to choose? The answer is, you need both, and there may be some things about the process that you do not like and that you cannot control, like what the most sucessful treatment will be. Let me clear up a real populuar myth for you, using medication and sitting on your thumbs are mutually excusive activities. Medication is a tool that many children need to make all the therapy, hard work, and intervention more effective, and the do a fabulous job of making kids with medical issues caused by neurological and developmental problems feel better while they undergo very difficult therapy. Let that go. None of what you are up against is an Either/Or situation. You do not choose public or private, you choose both, because that is appropriate, and you do not choose medication or therapy, because (chances are good) both are appropriate, and in any case, medication is never appropriate without therapy, although the inverse may be true for some.
You have some choices. You can either go see a Developmental Pediatrician or a psychiatrist/ Neuropsychological evaluation combo. Both will get you good evaluations that you can use to keep the schools honest. If you choose the latter, you may get in more quickly, but you will have to supplement with speech and language, OT, vision (developmental) ENT, Neurology, genetics, etc, and guess which ones of these are appropriate, and then try to figure out how they fit into your primary evaluation by yourself. I usually prefer a Developmental Pediatrician, because if you need ancellary evaluation, you get it in one full report. It also takes quiet a while, so you probably are 6 months out. You cannot speed up the process and get the kind of evaluation you need, even if you don;t want to wait.
Contact the school by writing them. Tell them that you suspect that he has a disablity, and you want him evaluated. Say that his educational need is (Blank) and list what he is having trouble with. They will contact you, and send you a copy of your rights. They can delay evaluation by meeting with you and telling you what in class interventions they will try first. You should document what they are going to do, and have a date that they will get back to you to tell you if these things worked. At that point, you can request the evaluation if he still needs it. Even if he has Asperger, and even if you have a private evaluation, the school must see an educational need to evaluate (thought you can inisist and win that round) and he must not only have a qualifying diagnosis, but also an educational need for seriveces to get special education accomodations and assitance. Being high functioning makes this a frustrating road. Log on to www.wrightslaw.com and start reading about advocacy, you will need to learn everything you can to navigate the system. Remember, the schools are there to educate him, and if he has asperger, his issue straddles the medical-educational fence, so you will be providing a great deal of his therapy and treatment, and the school will be providing educational serivices such that he is functional in the classroom. Anything beyond functional, falls on you, and that is the probelm with kids with apsperger, they are so bright that they seem more functional than they are.
Out put is generally their issue, figuring out how to tell the teacher what they know in just the crazy way that the teacher set up the assingment, even though they know the matieral better than most. If you can;t write the answer in a cloud, the issue becomes how well you can write in a squigly circle, and not if you know how to spell all the weather words you just learned. He will benefit from considerable accomodations, and you may find that they don;t always go smoothly, but the older he gets, the more important it will become.
If you find it overwhelming, you can find an advocate on the wrightslaw site, they will go to meetings with you, and help you navigate, that is what I do.
Never know less than the school does about your son. Know exactly what he needs, from the best private evaluation you can get, then negotiate for everything you can get him at school, and provide the rest privately. Get him the very best medical care you can, and don't prejudge what is out there, it is really nothing like what you are afraid of. Everything you try will not help, but something may work wonders.