As was said, in nearly all states, neuropsychology isn't a protected area of specialty practice. Any licensed psychologist can legally offer all services available to psychologists, depending on the state (e.g., some states may require additional training for certain things like hypnosis). It's typically up to each individual psychologist to decide what they're competent to do. I believe the same is usually true for physicians.
I know of two exceptions, although can only ever remember one: Louisiana. The second may be...Minnesota?
1) Maine is another one for neuropsych.
2) Other organizations are recognized in other states (e.g. HSPP has some special status in Indiana).
Can this person say they "specialize" in NP assessment?
Also, can non-neuropsychologists offer neuropsychological assessment as a service at all?
1) So long as they are not drastically misleading the public, they can say almost anything. It's a thin line between specializing in doing something, and having specialized training in something (e.g., I specialize in being a jerk, but I'm not a jerk-psychologist).
2) Maybe you are approaching this from a bad perspective. Our practice, and the practice of other professions, are governed by licensing laws. Our professions' licensing laws have no legal authority, or standing, to dictate how other professions practice. No outside licensing agency is going to change their laws because we complain.
a. Physicians can purchase neuropsychological tests because the test manufacturers say so.
b. Physicians can bill for neuropsychological testing because the CPT codes say they can.
3) After they've gotten licensed, I wouldn't try to gatekeep
a. You won't win.
b. It doesn't affect you financially, because there is more business than anyone can handle. Have you ever heard of an area where neuropsychologists' don't have a wait list?
c. It reduces your hassles. There are always going to be some quacks. And some difficult patients are going to be drawn to them. You don't want those patients.
d. Half of it is really funny.
e. The other half ends up with the patient being referred to someone competent.
f. Our profession has decided that the proper way to handle such issues, is to act like it's junior high, and make fun of people behind their back.
g. The "secret" of neuropsychology, is that it is built on a bad foundation. If the oldies could become competent without a fellowship, then fellowship wasn't truly necessary. If the oldies couldn't become competent without fellowship, then the first neuropsychologist was a time traveler. If you start raising the question of who had a fellowship, and who didn't, then the field falls apart. But I hated fellowship.
