It depends on what kind of report you are talking about. If you are talking about the "started at 11am, ended at 11pm, took xyz breaks, so and so was late" type of reports that might be needed for union or management documentation that is one thing. A completely different type of report is the "door broke, fence needs to painted, SQGG has a glitch, etc" type of report.
I take the first type of report during techs for documentation of hours, etc. The second type of report depends on the theatre. Sometimes the heads of each dept takes the notes at the end of each night, or I will send emails about any notes that someone who wasn't at the end of rehearsal might have missed. Another theatre I worked at the Production Manager maintained a list of notes that was kept in a public computer folder. At the end of each night, items that had been completed were removed from the list, and items that still needed to be completed remained on the list. New items were added. It was actually a pretty effective system, since everyone was looking at the same "to-do" list, and there wasn't duplication or things slipping through the cracks.
I guess I am saying what everyone else has said. It just depends on the situation...