As suggested, you need a voltmeter for the full picture. Finding a ground can be problematic for the voltmeter or the led's.
I suggest finding a good ground first. There might be one near the light fixture or you can tap the light fixture ground.
Once you have a ground then you can use a temporary connection with a thumbtack to pierce the wires associated with the switch. A simple 12 volt test light will accomplish the same thing.
Most likely one side of the switch is 12 volts at all times and the switch simply passes that through. Without pictures it is more difficult of course.