The logistics of peer review are generally simple, but they do require some forethought. If you want students to read papers in a round-robin exercise or to exchange papers with one other student, you don't need to require any photocopying. But if you want each student to read three other papers, make sure you remind students to bring three copies of their papers to class on the day of the exchange.
You can let students pick their own peer-review partners or group members, but you might also consider assigning peer reviewers based on your knowledge of students' writing and editing skills.
If you hold in-class peer-review sessions, circulate during the session to make sure students are on track and to intervene as necessary. Also, save a few minutes at the end of the session to discussion common problems with the class as a whole.