
Rex R. answered 04/27/19
Writer, public speaker, and visual storyteller
As you may already know, this is usually the responsibility of the Script Supervisor, with other departments keeping detailed notes for continuity as well (e.g., Costume Department noting the clothes the actors should be wearing if scenes are being shot out of sequence).
If possible, the best person to act as a script supervisor on a small shoot is the person who will be doing the editing. After all, if continuity issues arise during the shoot, it's the editor who will ultimately have to deal with them. But maybe that's not feasible, or maybe the director will also be editing.
In that case, have the director keep a hardcopy of the script on hand (not on a phone or other device). Write continuity needs on one margin of the script before the shoot starts, and try to think of everything that might arise during the shoot. While shooting, check your annotated script before each new shot. If continuity issues arise during the shoot, write them on the script right away. This will then serve as a useful guide during post-production.
Good luck, and have a great shoot!