As someone has already pointed out, professionals with a doctoral degree in school/counseling/clinical psychology may become licensed as psychologists. Licensing requirements do vary from state-to-state with many states requiring completion of doctoral work at an APA accredited school, completion of an APA accredited internship, passing the EPPP (and state-specific tests, sometimes) as well as completion of a post-doc (again, I know that states vary somewhat). In most states, when you've jumped all these harrowing hoops, you're a "licensed psychologist".
The tricky thing with school psychology is that many, even those with doctoral degrees, don't become licensed as psychologists; rather then are credentialed by the state board of education, or in some cases licensed through the psychology board (with indication that the license is specific to school psychology and not a licensed psychologist). The reason this happens is that it is not a requirement for school psychologists to be licensed psychologists; they are required to be appropriately credentialed to work as school psychologists. Interestingly, being a licensed psychologist does not qualify someone to work in the school setting-- they would still have to get the appropriate credentials as a school psychologist to practice in a school setting.
School psychology is a master's-dominated field; as another poster pointed out, only about half of those with doctoral degrees in school psychology actually work in school settings-- it's just not necessary to have the doctoral degree in most cases, unless you want to run something (i.e., special ed department, psych services, etc.), though there is a preference for doctoral level, if a district can afford it. In some ways, use of the term "school psychologist" is somewhat misleading because of the fact that most school psychologists are master's level clinicians (who are well-trained to provide testing services and counseling services specific to educational settings and within parameters of school-based/special education law, etc.). This is a debate that has raged in the school psychology world for years. Outside of school psych, the term psychologist is reserved only for doctoral level practitioners, and many doctoral level school psychologists agree with this-- but master's level clinicians generally do not. But as this thread points out, it really creates a lot of confusion for people who are school psych.
Being a master's level school psychologist is a good gig. The work is fun, the pay is good, the hours are great; it can be challenging and exciting, and very rewarding to be a part of the school community and a specialist in that community. In most states, however, being credentialed as a "school psychologist" (master's level) is going to allow you only to practice in school settings. Master's level school psychology trains specialists to do a specific school-based job. The job prospects are generally very good, also.
There are some doctoral level school psychology programs that essentially are master's programs with a dissertation (or some "similar" project) tacked on at the end. However, there are many doctoral level school psychology programs that train people to be viable as a licensed psychologist in various settings. If you are going to go to a doctoral program in school psychology, you should take a look at (or ask about) things like this: where do graduates work: only school settings? Or is there variation? What type of practica are available? Programs in or near major cities will offer much more varied practica sites. Are their faculty licensed psychologists (not just credentialed school psychologists)? Do they align equally (or more heavily) with the APA than the NASP? Does their coursework allow for a breadth of training, including in psychotherapeutic techniques, psychopharmacology, evidence-based practice, etc.? What are the options for tailoring your program (i.e., adding coursework of particular interest to you)? Where do their interns match? Do they even encourage their interns to go through the APPIC match (some school psych programs don't, but the good ones will require it)? By the way, there are some absolutely fabulous school-based APA accredited internship sites across the country, and clinical and counseling psych applicants try to get into these too. They're not typically as competitive as other types of sites but some are quite good. Make sure the program is APA accredited-- it is a huge limitation to end up graduating from a non-accredited program (e.g., much less likely to match, may not get licensed). Note that only doctoral programs are APA accredited; master's level programs in school psychology should be NASP approved. If it's not NASP approved, I'd steer clear. This can also prevent you from getting credentialed in some states, or at least make it a whole lot more difficult.
My master's degree is in school psych; I got my state credential (and national credential) and practiced in public schools. I loved it. But I wanted to be able to have more flexibility in terms of settings. I chose a school psychology PhD program in a major city near a major medical center, and I did all my practica in the hospital setting. I worked on national, clinical research grants; did some publishing; made sure to present at state, national, and international conferences throughout my time there. When it came time for internship, I matched to my top choice site-- which by the way is not school-based but is a clinical setting, where all the other interns are from clinical psych programs. My post-doctoral fellowship is in a clinical setting, and I have been approached regarding job prospects and offers ranging from academic to clinical to government positions. My program is top-notch, but I researched it carefully and chose it wisely. I also decided from the start what type of experience I wanted to have, built an area of expertise purposefully, and I am thankful on a daily basis to have not only the knowledge/experience to become a licensed psychologist but also to have the context and perspective of a school psychologist.
So I guess what I am saying is-- there are lots of things to consider in making your decision, and there is also some confusion about the field of school psych--- and rightly so, because in many ways school psychology has a hard time all getting on the same page together. I know many many outstanding and brilliant PhDs/PsyDs in clinical and counseling psychology who I respect. But I could not be any happier with the fact that my PhD is in school psychology.
Best luck to you on your decision.