Dancers learn steps almost always by having people demonstrate. The people from the Balanchine and Robbins Trusts, who are sent to stage works, are often called repetiteurs. (I've never seen "repetituese" in a program.)
In Russia, most, if not all, of the dancers, at least the promising and established ones, have a mentor, who coaches them in their roles. For example, in the "Sacred Stage" video, you can see Yulia Makhalina being coached by her mentor, and then coaching a promising newbie.
For a work brand new to a company, unless there is someone in the company who knows the work -- for example, Peter Boal has staged a number of works at PNB that he danced -- someone is hired from the outside. That person (or people) teaches the ballet. Often second and third casts, schedules permitting, will watch these rehearsals and learn from the sides.
Sometimes a choreographer, like Maillot for his "Romeo et Juliette" this year, will send a stager(s) ahead of time, and then will come to work further or fine tune the work.
Another alternative is that a company may have done a piece before, and an internal Ballet Master will set the corps and demi-soloist work, but some of the Principals may have additional coaching sessions. For example, Suzanne Farrell was in Seattle for a few days a couple of years ago, and worked with at least some of the Principals in "Diamonds".
Each stager works differently. Some have muscle memories like elephants. Francia Russell relies upon her notes. A number of stagers who have been at Q&A's, like Russell who used to moderate them, are wary of using video as much more than a memory aid, because they to a last person have said that mistakes creep in, and there's too much risk in following a video blindly.
Not many stagers seem to rely on notation, which is often remarked upon as being more appropriate for modern dance, but Doug Fullingon is an expert in Stepanov notation, and has done beautiful reconstructions from them.
Ballet biographies are full of descriptions of being taught from one dancer to another, from Primas who sought out Karsavina, to corps members who teach each other the roles just before the curtain goes up to cover for injured colleagues.
For re-working, often the company Ballet Masters are responsible for teaching and re-teaching, and there's a lot of peer-to-peer teaching as well. Of course, when there's a Balanchine around, and no one remembers the steps, he might just toss the old version and make a new one

In nearly every pre- or post-performance Q&A I've seen, this question comes up. So you're not alone in wondering.
I'm sure some of the people with professional experience could comment further on their experiences.