each dept is responsible for their own continuity - such as make-up, hair, props, set etc
standby props is a job where someone will be on set throughout filming, resetting things between takes such as pints of beer, cigarettes, opened letters, etc - they replace and reset anything that gets moved or damaged or used-up during a scene.
the continuity person will spot an error on the monitor and inform the relevant crew member to deal with it.
although i have known a few continuity people who were more like rainman with their attention to every dept continuity
script supervisors tend to focus on the larger things like position, eyelines etc and noting where things and people are and what was said when etc