Duty personnel were the personnel on duty at their particular duty stations.
The signage on the doors of the Earth Cargo Service freighter ECS Fortunate features guidelines how to react in emergency situations. It says that in an event of pressure loss all duty personnel should take on pressure suits. (ENT: "Fortunate Son")
While the USS Enterprise was in orbit of Psi 2000, Lieutenant Uhura made an announcement signaling for the engine room to go on standby alert, and for all duty personnel to report to the Bridge. (TOS: "The Naked Time")
Captain James T. Kirk put all duty personnel on yellow alert when it was discovered that an Orion scout ship was pacing the Enterprise as it transported a delegation to the Babel Conference in 2268. (TOS: "Journey to Babel")
In 2269, Captain Kirk put all duty personnel on yellow alert upon entering the Delta Triangle. (TAS: "The Time Trap")