First, you need to figure out which wire is which.
Then, you will wire it as follows:
Before you begin, shut off power at the breaker or fuse box, and then remove all the wire nuts inside that box.
Connect all the ground (bare/green) wires together, and ground the switch.
Connect all the neutral (white) wires together.
If you look at the switch, there will be a terminal that is labeled "common", "main", or "hot". Connect your power supply wire (black) to this terminal.
Connect the remaining each of the remaining black wires (each one should go to one of the lights) to each of the two remaining terminals on the double switch (in my photo, those are the terminals on the left side, clearly visible).
Mount the new switch, turn on the breaker, and voila! Separate controls!
I think that you should also use a voltmeter set to continuity and a long piece of wire to determine if each light is it's own wire, or not. Attach on lead to the long wire, and connect it to the center contact on the light bulb socket of one of the lights. Then connect the other lead to each hot wire coming in to the switch box. You should be able to figure out using this method if each light is its own wire, and then which each light's wire is.