Air carrier flight crew reported loss of aircraft control during pushback from gate.

Narrative

After pushback onto the line; the push crew disconnected the aircraft tow bar and comms without notifying the captain. They gave no set brakes" call and without the Captain (CA) confirming "brakes set; cleared to disconnect". The aircraft began a slow roll forward and myself and the CA assumed we were still being towed. The guide man alerted us by waiving his wands and giving us the stop sign. I noticed this and immediately said STOP to the CA. He stopped the aircraft and asked the push crew why no set brakes call was made; they disconnected before answering. After establishing comms again; the push crew apologized and admitted they never gave a set brakes call and that they were never cleared off. The push crew apologized again and admitted their error. After a pin and a salute; they drove back in the tug to the gate and the flight departed without incident. Cause: Lack of use of established procedures and challenge and response items by the push crew/tug driver. Suggestion: A reminder to the push crews that adherence to procedures; proper communication and terminology are essential to a safe operation."