In teaching Econ 560 (Topics in International Economics) remotely last spring I found that the breakout rooms worked very well in keeping the students engaged. A few notes about using them that I found important are
1. You need to give the students detailed instructions of what they should accomplish while in the room. You can post those detailed instructions in the “chat” window so that all students in all rooms can see them.
2. It is helpful to assign a student at random to report out what conclusions/answers the students reached in the breakout room. I use birthdays to designate the reporter (e.g., the earliest birthday in the calendar year, or the nearest birthday to April 15), and have found the students able to work with that.
3. From student feedback, I get the impression that there is swiftly declining marginal productivity of teams in breakout rooms. Limiting work there to 3-5 minutes, but doing it multiple times in the class period, seemed to work best.
I second everything Pat said, but would add that students seemed to appreciate being in the same group each class. I got more engagement from students when they felt more comfortable with their group members. For my upper division class in the Fall I plan to put them in groups the first week and maybe change groups 1-2x during the semester so that students can feel comfortable, but also meet more than 3 of their classmates.