Serious skating isn't just about what happens on the ice. There are all sorts of vital related issues: what to wear, scheduling, equipment issues, and more. When do these things get discussed, and do you have to pay for the coach's time?
I don't know how it is in other districts, but in Chicago pay for classes ranges from minimum wage (yes, you heard that right) to no more than $32 per hour; in other words a fraction of what the coaches charge for private lessons (Chicago range is $17 to more than $100 per half-hour lesson). Some, but not all rinks pay for a small portion of non-ice time for talking to parents, planning curriculum, etc.
Therefore, I would say respect the coach's time and keep these discussions to a minimum. Expecting a class-only coach to stay for more than 5 or so minutes after a class to discuss your personal skating needs is not fair to the coach. If you're just taking classes, your issues will not be so complex that they cannot be dealt with either on the ice or in brief conversations afterwards. If your issues become complex, then hire the coach to do private lessons.
Likewise, when the class ends, you need to let the coach leave the ice. The pay stops when the clock says the class is over, even if there is empty ice right afterwards. Maybe the coach stays after sometimes, but this is out of the goodness of his heart; don't exploit this. Remember that the coach cannot leave until you leave, because of insurance and safety regulations. He's not hanging around because he loves watching you skate without getting paid for it. He's hanging around because he's not allowed to leave until you do, and he's too nice to tell you to please go home now.
The other thing to remember about the class coach is that she or he is obligated to everyone in the class-- taking up his or her time during class with your personal issues is essentially stealing time from everyone else in the class. If you do it after class, but everyone else wants to also, then that coach's five minutes for which they aren't paying suddenly becomes, in a class of 10, nearly an hour. Trust me, it has happened. I keep business cards in my pocket and ask people to email me. You can also ask a class coach if they will set aside a time when you can talk to them.
Here's what you should NOT do: Do not grill the class coach about competitions, progress, private lessons, and USFS rules, and then hire someone else for privates. It's one thing to interview coaches to see who is the best fit, and we are all happy to give you as much time as we can- that's just marketing. But it's just exploitative to get all your information from the accessible class coach, and then turn around and hire the one who wouldn't talk to you without getting paid.
Here's what the coach should not do: Leave his co-teacher on the ice dealing with the kids, while he spends his time talking up the parents and soliciting private lessons. Do you want the coach who is teaching, or the one in the lobby schmoozing you. Just sayin'.
Like choir directors and university professors, your private coach pretty much expects to work on your behalf during non-lesson time. This is one of the reasons the private fee is higher than the class fee-- the coach's obligation to the student does not end with the clock. I estimate that I spend at least a half-hour off-ice for every 2-3 hours on ice on each student, working on choreography, schedules, parent education, etc. Coaches whose students compete or test have a much higher ratio. The private coach is expecting to converse off the ice, exchange emails, and talk on the phone. But again, don't abuse this. Don't demand daily hour-long conversations, or email exchanges that go into several daily iterations, or weekly off-ice meetings. Look at what your or your skater's actual "career" (such as it is) involves. The fewer the number of lessons, the less "competitive" (as opposed to recreational) the student is, the less you should be taking up the coach's private time.
One thing you can ask from private coaches is periodic goal-setting and benchmarking meetings. If you're competitive, you need one at the end of the season, a couple during early pre-season and one before each major competition. Some coaches will charge for these, some will not. For non-competitive students, twice a year or when either the coach or the parents feel a need for instance, if you want to switch a student from recreational to competitive, or when they reach a milestone like the axel that might require more ice. For situations like these, you don't want to feel pressured or distracted about missing instruction time.
If the coach does not charge you for these, and you feel that she's gone above and beyond, think about giving her a holiday bonus at the end of the year, like you do for other private contract employees like housekeepers, gardening service, etc.
In any case, never call the coach at home during what a reasonable person would consider personal hours-- weekends, after 9 at night, et cetera, unless you've been specifically told that this is okay. Use the phone number and the email the coach gives you; don't track down alternate ways of contacting her. Coaching is a business; keeping a business-like approach, honoring everyone's time, money, and personal boundaries always makes for a better coaching relationship.