Your question is short and to the point. The answer is a bit more complicated.
In most countries, to practice counseling or therapy, a professional counselor has to go through extensive education and training. Many countries have a licensing process as well. Trained, licensed professionals work in both kinds of settings. The choice clinicians make about where they work has to do with their personal philosophy about counseling and their own career goals. Many counselors begin their careers in a public mental health clinic and then go on to private practice, it’s true. For that reason, it’s often the case that a private practitioner has more experience. But it’s also true that there are many highly experienced, caring therapists who prefer to work in a public clinic where they have more interaction with other professionals and opportunity to work with people who might not be able to afford a private clinician’s rates. The setting doesn’t determine whether you will get the best care. The quality of the individual professional does.
Choosing a therapist can seem like a risky business, I know. After all, you are choosing someone who will get to know your secrets and your pain as well as the parts of you that you normally present to others. It’s fine to ask about the counselor’s credentials (education, years of experience, commitment to staying up to date with research, etc.) as part of the initial interview or intake. Professionals don’t take offense at being asked a legitimate question about their qualifications. You can also confer with your doctor about her or his patients’ experience with that particular counselor or clinic.
I wish you well.