As you can see, the circuit consists of two switches, two LEDs, and a resistor in series (and yes, they actually are in series - there are no hidden wires). So how does each switch control one LED? Solution after the break:
Once I got access to a pen and paper, I figured out how the circuit worked (or at least one possible solution). I then rebuilt the circuit using a spare cell phone charger and components I had lying around.
Here's a schematic:
For the circuit to work, it needs an AC current, rather than DC as this cell phone charger produced. To get an AC current, I opened up the charger, and moved the output wires so that they bypassed the rectifier after the step down transformer. At this point, the charger puts out 5V AC.
Here you can see where I hid the diodes. The diodes under the 10mm LEDs are tiny schottky diodes I pulled from a printer power supply.
Diode across each LED:
Diode across each switch: