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The light sockets and switches are in series. When both switches are pressed, the circuit works as expected and both lights turn on. When only one switch is pressed, only the light across from it comes on.
The circuit relies upon alternating current and four diodes across both switches and both sockets. The diodes are oriented in such a way that current will flow clockwise when the right switch is pressed and counter-clockwise when the left switch is pressed.
This circuit appeared in a Martin Gardner article many years ago. It is possible to open the base of the bulbs and put the diode in the space between the glass bulb tip and the metal base so that one switch always turns on one bulb regardless of which socket it is in.