That they do. Staff have a CA and PA just like players, and get better over time. I'm not sure how complex it is, but I'd assume their mental attributes and the facilities they have access to would affect their growth.
The main thing that effects their growth is their experience (So if they work at a top club for a long time they will grow more than if they were at a lower club) and going on courses (To get licences). This is especially shown with players who retire and become staff. Alot of them have very poor staff attributes, but most of the time they will quickly grow.
I have Scholes, (Phil) Neville and Henry as youth coaches, as although there attributes are quite low there is a good chance they will become good coaches