Everyone must do his or her part, and do it on time. While group members usually want to give other members every chance to participate, the group should never delay its work because a non-participating member missed a deadline. Make the deadlines clear and then move ahead without the procrastinators. Find an equitable way to share the work. There are a number a ways to do this.
Some groups alternate responsibilities on each cycle so that everyone takes on different roles.
Your group can work out its own process. However, it is extremely important to discuss this before you embark on a review cycle and periodically as the class continues. Try not to take things personally. One of the problems with email it that it is easy to make assumptions about the other person's attitude and motivations since you have only the written word to rely on, and you do not get the visual cues that you rely on when talking to someone in person. If someone writes something that upsets you, write back calmly and ask if you understood them correctly.