At my current gym, we have a book that sort of says when we have privates and so forth.
I've worked at gyms, where what I was paid at was simply my hourly rate, which IMO really sucked. My hourly pay rate was decent but for a jr coach that was making half what I was making means the gym was keeping 75 to 80 percent of what was being charged. That's unheard of in the PT field even at globogyms. I still enjoyed doing PL but I tended to only do them if I got along with the student and family and if they was a good reason for them and if they were going to work hard.
I've also worked at gyms where the rate was set by the gym and we recieved a cut. I can't actually remember that cut right now but I wasn't being hosed as much as the above gym. I want to say it was around 50-60/hr and 25-30 for a half hr.
Personally, I prefer it that the gym gets a per hour fee of 5-10/hr and I can charge whatever I want. In the past I didn't charge as much because I charged it relative to what I was making ( aka 10/hr ) so 25-35/hr was a big increase even if I payed the gym 5 or 10 dollars. Some gyms didn't charge the coaches anything for PL which is nice.
From what I've heard typical cut is 70 to the trainer, 30 to the gym to up to 85 for the trainer and 15 for the gym. Some globogyms run at percentages of 45-65 for the trainer though I have heard of clubs that pay personal trainers only 10-15/hr.
I don't mind groups of 2 as long as they will work well together. This becomes difficult if their skill level is too different. It also allows each gymnast to get a bit of a break in between. Even 3 can work out well, but anymore should just be a class. Having 2 or 3 come in for an hour instead of 1/2 makes a lot more sense.
Personally, I think the parent/child should be allowed to pick the coach. This only becomes an issue if that coach teaches something drastically different than their regular coach. This can be an issue if a parent/child is going to a different gym for PL than the gym they go to gym at. Sometimes, I just didn't have the schedule to be available for privates on certain days or at certain times.
I try to see to it that the gymnasts coming in for privates do their own warmup, one that I will typically show them so we don't have to utilize the PL time for that.