I agree with with pennyroyal said. In the 20 some years that I've been in psych nursing, nearly all of the CNSs I've known have had private, outpatient psychotherapy practices -- that used to be the main reason for becoming a psych CNS. When I went to grad school 11 years ago, I was considered quite an oddity because I prefered working in acute, inpatient settings -- all my classmates intended to be outpatient therapists in private practice.
The CNSs I have known in private, independent practice typically have had an informal relationship with at least one psychiatrist, to whom they would refer for medication (when indicated) and if insurance companies required that the person be seen and referred by a psychiatrist before they would pay for therapy. The psychiatrists also referred clients to them for therapy, but they were not "employed" by the psychiatrists.
Now that Rx authority has become an option for psych CNSs in so many states (not mine, thank goodness!), psych CNSs in those states are mainly employable for prescribing and med management. I interviewed a few years ago for a position in the state next to me and was told that, because CNSs can get Rx authority in that state, I would be expected to get Rx authority ASAP and be utilized by the clinic to see people for 15 minute medication appointments. I was also told that no one in that state would be interested in or willing to hire me as a psychotherapist, because my time was so much more valuable writing scrips. Of course, if you have your own practice, it is entirely up to you how much time you spend doing therapy and how much time you spend pushing pills.
The psychiatrists I've known over the years who have employed nurses have generally employed NPs to do H&Ps and med management. They have not employed CNSs as therapists, because you can hire social workers and LPCs as therapists for a lot less money.
Over the years I've been a (child) psych CNS, I've worked in acute inpatient settings, outpatient clinics (community mental health centers, as an employee), and have done a very limited amount (by my choice) of independent outpatient psychotherapy. If I wanted to do more of that, I would certainly be able to.