I think a good DP, as well as a good anybody on set, should have a basic idea of how the entire process works. You should be aware of everyone's job and have a good idea of what they do, how they do, and how well it works for them and for the show.
From a technical standpoint the DP should always be doing his/her homework and learning the in's and out's of his/her craft. He/She should know how to achive what he/she needs to on set, this allows for better communication between the DP and the gaffer or the DP and the AC.
Everybody has a job to do on set, and the DP's job is to communicate effectivly to get the results that the production requires in order to keep the day moving along.
BTW I work mostly as an AC when I'm not DP'ing, but Lighting the set and knowing what my Gaffer is doing and learning all I can about how to manipulate the light is what really gets me excited about the work.