Because the private college counseling industry is unmoderated, meaning that no one organization sets standards for everyone, there really is no one certain degree one must have to be a private college counselor. However, that being said, most well known counselors (ex. Dr. Cohen, Dr. Hernandez) have degrees in something. Dr. Cohen has a pHD in Latin American literature, Dr. Hernandez has a pHD in education. However, you can see there is quite a difference in the two degrees.
Dr. Cohen and her staff at IvyWise all have or are obtaining "certificates in college counseling" from the UCLA Extension School. This program is the first and the only program of its kind. Take a look at the following for more info: www.uclaextension.edu/collegecounseling
It's also important to note that some private college counselors have worked in an admissions office at a university at one time or another. Those who have worked at highly selective schools can advertise this and it convinces people that they know what they are doing. Using the example of the two mentioned earlier, Cohen has worked at Yale, while Hernandez has worked at Dartmouth. Catherine Cohen says that "IvyWise" is a "state of mind...If you know what the top schools want, you know what all the schools want."
College Confidential is a internet community, but it is also a private college counseling service. Check out their main website and see what they have to offer and what qualifies them to offer their services.