Everyone is responsible for his or her own time. The key is agreeing on the time. Everyone has to be in agreement on how the music will feel. Also, if you're outnumbered, you have to be open enough to go with the others. I think the concept of a rhythm section agreeing on the time is not so much a verbal understanding discussed between players, as it is an intuitive one.
Musicians are varied, but time as a concept is a constant.
Some players play on top of the beat. Some players play on the back side while others play right on. I believe there is no right or wrong here, just human interpretation. What you look for in a good rhythm section is similarity in everyone's internal clock. In the future, I will be interviewing more rhythm section players in regard to their concept of the good bassist.