A319 First Officer reported that SOP's were not followed during gate push-back. Communication with the ground push-back crew was not clear; distraction from another aircraft passing near and CRM procedures not followed resulted in brakes not set during engine start and the aircraft moving several feet.

Narrative

During pushback from gate X; ground told air carrier X that 'normally our company stops the push much shorter.' Air carrier X was parked behind us and off our right wing; a shorter push would have allowed them space to get by us. Ground then called and asked if we had communication access to our push crew; to which I answered yes. At this point the push crew had us stopped at the point they intended to end the push. Shortly after ground asked us to push deeper; and I relayed to the Captain who relayed to the push crew. In piecing this together after the fact (since I was on ramp and the captain on intercom with the push crew) we believe right before the request to continue the push deeper from captain to ground crew; the ground crew said 'set brakes.' The Captain did not give 'brakes set pressure normal' verbal to the push crew rather asked them to continue the push deeper. They pushed us a few more feet and then ground called me and said to just stay where we were; which I passed along to the Captain and he passed along to the ground crew. At some point after; the captain said to start both engines - I started the first and before going in to the second start our attention was diverted to air carrier X squeezing through a tight spot off our right wing. The Captain released the crew to disconnect; we received a salute; and the 2nd engine was started. Towards the end of the start; the aircraft started rolling forward; perhaps a few feet at most. We noticed the movement and the Captain immediately brought us to a stop and set the parking brake. The Ground Crew was well clear of the aircraft during movement.