A commenter gave a great insight from a college PE pedagogy class: "everyone always moves." It's a terrific piece of advice for a sports program. After all, you're not there to write term papers. You're trying to learn how to move!
Of course, this does not mean that you can move in any random pattern, according to your own desires, and whenever or wherever you feel like it. There are always issues of safety, courtesy, and space.
The commenter goes on to talk about a teacher who said yes, but you have to limit activity on a given apparatus, for safety reasons. And this is the crux of the matter. In an ice rink, the entire room is our apparatus. And like other apparatuses- whether a balance beam or a baseball diamond- there are rules about sharing and appropriate use.
Where to skate
In class, you've probably got a restricted area for the students in your level or class only. Unless the teachers give explicit permission to cross into another level's ice, you stay in your own boundaries. (Skating through another class is fairly common in freestyle classes, unheard of in Learn to Skate.) In private lessons you skate where the coach tells you, or according to whatever set pattern the rink has laid out.
When to skate
When the coach says "go." Period. This is not negotiable. It's really easy to move on skates, and kids (as well as a lot of adults) have ants in their pants-- they seem incapable of sitting still. I'd love to see how this works in an office, or a class room, or a car, or at random street crossings. If you've gotten stuck with a coach that talks too much, you still have to stand still. Next time, don't take class from that coach. But frankly coaches that talk too much are not really the problem; most of them don't. The problem is children who simply skate off at random times with no consideration for any of the above safety rules.
What to skate
The task at hand. This means that when the class is practicing spins, you should not be doing waltz jumps.
Who should skate
Everyone, or no one. If you are the only person moving, and have not been explicitly told "we are now going to demonstrate one at a time, and it's your turn" then You Are Doing It Wrong.
How do you know when skate?
Just going to take a wild stab at this, and suggest "listening."