As a general rule, licensed psychologists have a PhD or PsyD in psychology (you may find some exceptions due to grandfathering clauses when licensure was instituted for that state). Very research-heavy AND clinical training-heavy, 5-year doctoral degrees. I believe that psych doctoral programs do look for research on the applications, but I'm not the expert on that. The Psychology Forum would have the answers to those questions.
A licensed clinical social worker (LCSW, or whatever a particular state calls it) has a Masters of Social Work, while a licensed professional counselor (LPC, or again- whatever each state calls it) will have a Masters in Counseling, or MEd in Counseling, etc etc. And then the licensed marriage and family therapists (LMFTs)... These are all 2-year degrees. Although they require some research classes, the focus is on clinical training. Social work is the only one of those three fields that does not exclusively train their students in mental health. That is, you can get a MSW having done a concentration in Administration, Health, Children and Families, Mental Health, Community Organization, and many others. So as a general rule, having the MSW offers a bit more flexibility than the others should someone want to switch focus within the field.
All of the above practitioners are licensed to practice independently within their state.
Depending on how a practice is set up, it can be very autonomous, as is a physician's (psychiatrist's). I am the sole proprietor of my private practice. Many of my clients see psychiatrists for medication, and I do a lot of collaboration with those doctors when possible, but they do not dictate my practice style. They do send me referrals, though, so it behooves me to stay friendly with them. The collaboration also helps me do a CYA in the event of a negative outcome.