That's exactly right. You should be the only person on book. Everyone else should be present and ready to jump in. Of course, depending on the show, there may be long stretches where the bulk of the cast is not doing anything, so you need to (nicely) let them know to please just hang in and bring something quiet to do until it's their scenes — chatting and wandering off are not options and become very distracting.
Is it an Italian run (double time, quick cue pickup with no gaps between lines but reacting to each other) or just a super fast line run with no acting values at all, which is about whether or not you actually know your lines and cues? Either are legit, depending on what the director wants and what the cast needs.