At my theatre, at the start of tech everyone on run crew gets a master run list (created by the PA) that covers who does what when; whether the cue is given over headset, via cue light, visual, autofollow, etc.; the shift number of each move; and any notes. The run list covers moves made by actors of furniture during the action of play, behind the scenes shifts, prop hand-offs, everything. As tech progresses, it is each crew member's responsibility to take notes of what they do that changes, and turn their notes into the PA at the end of each night. The PA then updates the run sheet, redistributes, and the process starts again.
There are some shows where the run sheet has ended up being 20 pages long, which is a large amount of paper to have shoved in your back pocket during a show. I usually type up a personal run sheet in Excel that lists just the shifts that I am involved in, and add personal notes ("remember to X US to [actor] to prevent traffic jam", "You need to haul a** back on deck after this shift to make the next one...").
The PA has a master copy of the script backstage, but I've never seen it referenced during the course of the show; we only use run sheets.