With my Saturn, it should only try to go into pairing if you have the PDM button pressed before and during switch on. Both LEDs should light in normal operation to indicate a good battery. I have noticed that it doesn't like rechargeables - it complains of low battery by bleating at me, but is perfectly happy with alkalines. If the one that only shows one LED doesn't bleat, but works OK otherwise, its probably just the LED not working, rather than anything deeper in the system.
Totally non-intuitive binding -
1. Turn on the receiver power supply first, press the "pair" button for more than 2 seconds. The receivers LED will blink quickly.
2. Hold down the transmitters PDM button then switch on the transmitter power switch. The transmitter Red and Green LEDs will flash alternately, then release the PDM button.
3. Press the transmitter PDM button again.
When pairing is complete, the receiver LED will stop blinking and stay on.
The transmitter LEDs will also stop flashing.
4. After a few seconds, the receiver should start to respond to the transmitter.
5. Finally, turn off the transmitter and back on again to store the pairing.
Doing step 2 should indicate whether it is a dud LED or something driving it.