The Police Maverick appears when the player has a wanted level of 3 or more.
The helicopter has an "invisible" 5.56mm machine gun on the fuselage, as well as a searchlight and a loud speaker. In GTA San Andreas and GTA Liberty City Stories, the interior of the helicopter is empty and there is no visible pilot flying the Police Maverick when it is chasing the player. This problem (in GTA San Andreas) was rectified in GTA Liberty City Stories by simply having the windows of the helicopter blackened; obscuring the interior, but this was more likely to create the same effect as with its predecessor. Its windows aren't black when flown by the player, though they are if the player manages to steal it from the police by reaching it via cheat devices.
Because of such things, players believed that there is an actual ghost, who is flying the helicopter. The effect is likely just caused by a simple limitation, because it wasn't assumed that players would actively be staring into the cockpit, so the pedestrian was removed to prevent strain/glitches on older systems.