I would have thought that most chroegraphy is learnt and developed "on the job" rather than written down in advance and then given on paper to the learner. Ant
That has been my experience in musical theatre and modern dance. Especially for large groups, the choreographer may have already worked out the steps and patterns in advance and may have made written notes of some kind, but the process of communicating them to the dancers takes place on the floor of the rehearsal studio, not on paper.
(I remember one occasion when the choreographer had a broken hip and was on crutches so he couldn't demonstrate the choreography to the dancers himself. Once when he was inspired in the middle of the night, instead of writing down his ideas he woke up the dance captain, who was his roommate for the summer, to teach him dance through words and gestures, so that the dance captain could teach the rest of the company in rehearsal the next day.)
For dance companies that want to keep a record of a dance perhaps to recreate it with other dancers at a future date, I believe that many do use video records these days.
For competitive skating, the main point of the choreography is to showcase the skater's technical ability. Therefore, the choreographer's artistic vision may need to be changed or even compromised to allow the skater to execute the program to maximum competitive advantage, which is why there needs to be give-and-take on the ice with the skater and coach. And once the skater has learned the program, there's no need to write it down because no one else is going to skate that exact program again.
For artistic skating in context such as ice theatre, the choreographer's vision can take precedence. Different skaters may need to take over another skater's role in a performance, or even perform the same solo on a different occasion. So in those cases some sort of record of the choreographer's intention would be useful in communicating to the new skater. But again, it might be on video rather than on paper.
Here's an artistic (not competitive) work that was originally created for one skater and recreated 30 years later by another. The choreographer was not a skater, so the original performer did have a lot of input into the choreography. I haven't done a comparison of the two versions, but I know at least some of the choreography had to be mirror-imaged to accommodate the second skater's rotational preference:
http://www.youtube.com/watch?v=SXJqsoFwUic&feature=related
http://www.youtube.com/watch?v=0W4-HXL_bbU