I build my own caving lights using LEDs and I find LEDs will glow with a tiny amount of current providing you have reached what is called the forward voltage. For white leds this is about 3 volts, red about 1.2 volts. I have seen them start to glow at lowervoltages though.
LEDs are not designed to be reverse biased. In other words you shouldn't apply voltage the wrong way round (unless they're protected by special circuitry) otherwise they can be destroyed. If you're using a led bulb designed to work from 12v you'll be OK and they usually can take having reverse voltage even if they don't work one way round, also you won't need current limiting resistors. However if you use a plain led they will need a current limiting resistor otherwise they'll only last a nano second at 12v. You must be using led bulbs because they work without a resistor. For your application the leds will only need 10mA or so each. So for a plain LED a resistor of about 100 ohms would be called for. Without it the current would be several amps and would instantly blow the LED.