IMO it is difficult to get these privates unless you have access to the clientele. If you work for a language school, it is unethical to take students away from them.
Children and teens are a different story when you work for a school or university. Many of these students have parents that are interested in private tuition.
I have students on a waiting list; whereas, I have only ever had a couple of adults seriously inquire about private lessons and they were through Thai friends. Once they found out the rates, they quickly declined.