My coach's policies say I'm supposed to pay her monthly in advance to reserve my lesson spot, but I've always just paid her the day of, and that has been fine with her, but I've also never not paid or not shown up for a lesson without giving her ample notice. I think she prefers that parents of her younger students pay her in advance because while she's on the ice she doesn't want to have to go tracking everyone's parents down after a lesson to get paid. Makes perfect sense to me. I would also venture to say that most of the coaches at my rink prefer to be paid monthly - it just makes record keeping much simpler for them, and when you are self employed, simpler is better as far as record keeping goes.
Ages ago, at sort of the same rink but under much different management, I would pay the rink for my private lesson time, then you'd get a ticket (or tickets depending on how much time you bought) which you would give to the coach, and the coach would have to turn those in to get paid (minus the rink's commission) however there was a perpetual problem with those tickets getting lost or misplaced, coaches not getting paid for their time and getting mad because obviously the rink got paid, but they kept NO log of who the lesson time was being bought for, so it was very difficult to convince them to give the coaches credit for those missing tickets. The rink eventually moved onto a logbook system, coaches log their private lessons, students pay the coaches directly, and the coaches pay the rink their commission.
In many ways the logbook system is much easier, and it accommodates coaches from other rinks coming in to teach if they need to (their ice is shut down, competition coming up soon, whatever) so long as they have appropriate credentials (mainly proof of liability insurance, plus PSA, USFS, or ISI coaching membership) since all of the coaches are considered private contractors, not employees (most coaches also teach LTS though, and technically are employed by the rink for that, and I think it also reduces their commission rate, since they get paid much less than their private lesson rate to teach classes, so it's a bit of compensation). The front desk staff does somewhat monitor the logbook too, and if they notice a coach is teaching and hasn't logged anything, they remind them to mark down their time.