Am I correct in thinking both license lights are working, but the dash says "license plate lighting out" or words to that effect?
Just for fun I disconnected one of my lights and started the car, there was no indication of a problem until 10 seconds after I turned the lights on when the message came up. Shutting the car down and restarting I didn't get an error message again (2nd time) until 10 seconds after I turned the lights on. I believe this is different behavior than with a burnt out headlight or brake light. In that case the car checks the circuit continuously, but in this case it is only checking when the light is turned on.
I think all the car can check for is the current draw (amps) and in your case it thinks the amperage is too low when the lights are on. Assuming this is so, there are two reasons I can think of which could cause this: first the amperage is too low, and second the device in the BCM (body control module) which detects the current isn't working right. Short of replacing the BCM the only issue you can deal with is the first one.
My understanding is that in this case the BCM can be simplistically though of as a master set of relays which switch to allow current to flow to the various lights. There is a large 12 V feed wire to the BCM and multiple smaller wires which run to various power users and then to ground.
I think low current draw could be caused by corrosion on any one of many connectors between the current source (BCM) and the final ground connection. Are both the lights the same brightness? If not the dimmer one is likely causing the problem. Caper's solution above might be the simplest one - a higher wattage bulb might pull enough more amps to trick the BCM into thinking all is well.
If you want to find the cause of low amps I think you could check current flow by measuring the voltage drop between the positive battery terminal and the hot side of the bulbs and between the ground side of the bulbs and the negative battery terminal. I would try this by backpinning the connectors with the lights turned on and bulbs illuminated. You should be able to confirm that there is minimal drop between the positive terminal of the battery and the hot side of the bulb and between the ground side of the bulb and the negative terminal of the battery. If one of these is bigger than one volt then that is the circuit where I would start looking for corroded connectors, damaged wiring, or something else causing a voltage drop.
Having written that I think what a lot of work and can anyone reliably measure small voltage drops accurately enough without special tools? Maybe not. Probably just easier to look for corroded connectors in likely places - battery terminals, around the lights themselves, around the fusebox, the ground connector, the connector between the hatch and the rest of the car.