So first we need to differentiate whether you actually want to do "Concierge" or whether you just want to do "insurance free" because they are two very separate methods of practice.
Concierge generally requires a wealthy subset of patients. It comes in 2 flavors. The most common, like with MDVIP, is a yearly "Access Fee" that covers longer office visits, enhanced communication with the doctor (text, e-mail, whatever). Those doctors still bill insurance just like everyone else, they just see fewer patients/day and still make good money because of that Access Fee. The other type is no insurance but usually a higher rate (MDVIP for instance is around $2000/year while insurance-free concierge is most like $5000 and above/year). No insurance is required for that, but the significantly higher access fee makes it harder to attract patients.
Insurance free is very simple and comes in 2 forms as well. You can do regular fee-for-service, meaning patients pay you every time they come in. Your fee is usually lower than insurance-based offices since you need less staff to do it. Its amenable to walk ins and urgent care but you still have to have pretty good volume to make a living.
Subscription based is what I do. Patients pay a monthly membership fee (mine is $50/month) which gives them unlimited office visits, my personal e-mail/cell phone number, in office procedures, and certain injection medications. Its kinda like the more expensive type of concierge but much more affordable so you have a much larger pool of patients to draw from. The nice thing is that since patients pay whether or not they need you every month, most of my days aren't too busy. The most patients I've seen in a day since I opened was 12, and most days its around 6-8.