If the meeting is being cloud recorded, it will only record the main room, regardless of what room the meeting host is in. With local recording, participants can record the breakout room they are in. Multiple participants can record locally.
In the Recording settings, navigate to the Local Recording option and verify that the setting is enabled. Also, check to make sure "Hosts can give participants the permission to record locally" is enabled.
If the host has already opened the breakouts rooms, they can join your breakout room to allow a participant to record. Participants can also ask the host for permission to record.
Click Record in the meeting controls to start a local recording.
Click the pause or stop icon in the meeting controls to pause or stop the recording.