Private practices are on the way out in America as market forces are causing them to be swallowed up by hospitals and "networks". It is almost like the independently owned grocery stores that have been slowly eliminated from cities and towns across America by the rise of the corporate supermarkets.
The model, back in the day, was a solo practioner who bought or rented a space, furnished it, hired a nurse who also handled appointments, hired a book keeper to send the bills (or which he may have done himself -- almost always a "he") and an accountant to figure the tax bite. The OB who delivered me and the pediatrician who took care of me as a kid had this style of practice (1950s-60s). There was very little technology in the day and docs could do their own lab work. Another model was a partnership between two physicians and in some cases an older physicain would take on a younger partner who would eventually buy out the older partner. Finally, there is the partnership that hires associates who work for a salary while the partners are paid out of the profits (what they make can go up or down depending on how profitable the business is).
Fact is, most doctors became small businessmen, not because they loved business but out of necessity and Issues about billing, insurance, medical records, electromic medical records, HIPAA, labor law, hiring policies, and much more gets in the way of taking care of patients. Furthermore, if you have to buy the furnishings including the medical equipment (like all that equipment the eye doctor has you look in to check your eyes) can be a huge expense that ties up one's capital, or forces one to borrow. Some people would rather work for someone else on a salary than worry about the business side of the equation.
If you aren't working in a practice, then it really isn't your practice, is it? A person can open a clinic, administer it as a business and hire people (including physicians) to work in it. This isn't a "private practice" but more of a clinic model. These can be for-profit or not-for-profit. The person wouldn't have to be a physician and in some cases the training for such an endeavor is a MBA or MPH in health care administration, usually pursued after having worked in the health care industry for some time. In some cases the person in charge is a MD or RN with a master's degree in health care administration. TIt really can go either way. The fact remains that these clinics are also being bought out by hospitals unless they are serving the very poor, and therefore are not profitable, or operated as family planning /abortion clinics and in some cases already part of a federation (such as Planned Parenthod).