In terms of qualification there is no difference between a licensed speech/ language pathologist/ therapist who works for the Sd and one who doesn't. There's one national license exam. Each state can have different license requirements... eg # of hours of practical experience. But the masters program and the exam are the same.
The real difference in this profession is the specialty. Those who have spent many, many years in a school have limited their practice to children and to the tests that the SD wishes to do. As compared to the person who works for a hospital and does a lot of testing of adult stroke victims for swallowing and sound production. The particular professional who saved my child from more years of torturous speech therapy was a speech pathologist whose specialty was evaluating for oral motor function.
I've had a lot of experience with speech people re my Pervasive Developmental Disorder (PDD) son and my ADHD son. In all honesty, if there is a real concern about a "communication disorder" eg aphasia or apraxia, I'd take my child to another setting. There are actually much more extensive tests that can be done in a hospital setting... if that's necessary.
In answer to the question, yes, if the school person is licensed.... and I've never seen one working for a school that isn't... they can diagnosis any specch/ language problem. But whether or not they have the experience to do so is another question. Should you have a pediatrician operate on your child? After all, a doctor is a doctor.
This website has a lot of good articles. Also, cd.com off the home page has a lot of links and articles too.