It depends on the choreographer. I've worked with people who use a couple of different styles. Some listen to the music ahead of time, plan out the program, and come in ready to teach it to the skater. Depending on how quickly the skater picks it up, that might take one or two sessions. (I'm talking lower level skaters here; more intricate choreography for advanced skaters or for teams could take longer.) If the coach is a skater, they often demo, then have the skater shadow them, then have the skater try it on their own. Sometimes they write it down, listing the steps in order, and sometimes they draw it out, kind of like a dance pattern. It is expected that the skater remember from lesson to lesson, although, obviously, things get reviewed if they don't. Some will video themselves skating the program, so the skater can watch and learn that way.
Other choreographers work more organically, and create the program as they teach it. They watch how the skater does the first moves, and that suggests the next moves. This process often takes longer. When I've worked with choreographers who did it that way, they have been dancers, not skaters. I don't know whether that's what made the difference, or whether it was just their personal style of working. My Russian choreographers were dancers, and so couldn't do complete demos on the ice. They would tell me what they wanted me to do, and could show me poses or arm movements, but didn't skate at all. Sometimes they would physically move my body into position, if the language barrier failed us. In both cases, if the choreographer doesn't draw out a map of the program for me, I do it myself. That really helps me as a reference, but I learn kinetically rather than visually - through muscle memory, if you will. The map helps, and the demo helps, but I learn the choreography by how it feels as I execute the movements.
In both cases, an outside choreographer will often place the jumps and spins in the program, but leave the specifics of exactly which jump or spin to the skater's technical coach. Most choreographers I've worked with will modify the program as they go if the skater finds something particularly uncomfortable.
When I have choreographed, I generally work more in the first style. I come in with a program planned out and drawn out. It helps me remember, and helps me work more efficiently. If things get added or changed as we go, I alter my map. Eventually I give a copy of the map to the skater, but they can use it or not, depending on how they learn and remember. (Sometimes it's mostly for Mom to have for their scrapbook!) I work this way with classes, too. The few times I've had to plan a program for a class I wasn't going to be teaching myself, I gave the instructor a written plan and map and left them to their own devices to adjust it as necessary. For the purposes of a group ice show number, it wasn't important to me that my choreography be executed exactly as I had imagined it. It was much more important that the class looked organized and presented well in the show. If that meant the instructor had to change things, even a lot, well then, so be it.
In my ballet classes (adult, multi-level, including some beginners), it is routinely expected that we execute a combination of steps after seeing it demonstrated only once. As you build up a vocabulary of steps, this becomes easier and easier. The ability to pick up step sequences quickly is definitely something a professional skater or dancer is expected to have.