Air carrier flight crew reported the aircraft rolled for 20 feet after pushback before the crew realized ground personnel had disconnected the tug without coordinating with the flight crew.

Narrative

Pushback crew disconnected ground power without permission prior to pushback. APU picked up the load but then shutdown causing us to reprogram the FMS once power was restored. At the start of the push; the Captain called Brakes set; cleared to push". Ground crew said "OK "and seemed to start the push. Captain told them they needed to tell him if they were ready for brake release. They said they were and to release the brakes. We pushed and were cleared to start engines. While completing the after start flow the aircraft started to roll forward. We were heads down and didn't notice immediately. After approximately 20 feet of movement the Captain noticed and applied brakes. Ground crew never alerted us the plane was rolling. They never told us the push was complete and to set brakes. They had already disconnected the tug and were driving it away. Captain called brakes set and the ground person said something to the effect of have a good flight and left.[The cause was] poorly trained ground crew and ineffective communication.[I suggest to] ensure ground crew is properly trained and as a crew; exercise vigilance that critical steps are performed."