Seriously, pilots almost never say over. I'm not exactly sure when this stopped, but I never used over at the end of my transmissions.
Usually pilots start every request with who they're calling and their callsign, either full or short (on the West coast of the US you can usually get away with just your make and last three characters after your initial transmission to a controller. If they are getting instructions or information from a controller they usually end their acknowledgement with their callsign. Examples:
(Cessna 182) N12345: "Seattle Center, Cessna one two three four five, with you, level four thousand"
Seattle Center: "Cessna one two three four five, radar contact, local altimeter two niner niner two"
N12345: "two niner niner two, Cessna three four five"
Seattle Center: "Delta six seven eight, descend and maintain flight level two two zero"
(737) Delta 678: "Down to two two zero, Delta six seven eight"
The real question is: Can this be used to eliminate that terrible noise that happens when two users try to transmit on the same frequency at the same time?