The connector on the Drivers side/left side/Port side has two black wires going in on pins 1 and 2. On top of the connector there are small holes over each wire. This is so you can test voltage on those wires.
My guess is that with the backup lights on, you will have 12V on pin #1 but not on pin #2 (the black wire) next to it. As Art B said, there is a wide piece of copper on the circuit board, that copper can get warm and deform enough to mess up the connection going to the Passenger side. If that is the case. You can strip insulation off the two black wires and electrically connect them. There is no fear in messing up the Functionality of the Bulb Failure Sensor. Backup lights are not part of that circuit.
The BB e-mail is still not functioning or I would e-mail you the color coding of the Rear tail light wires. I too am assuming you have a sedan , not a wagon.

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'75 Jeep CJ5 345Hp ChevyPwrd, two motorcycles, '85 Pickup: The '89 Volvo is the newest vehicle I own. it wasn't Volvos safety , it was Longevity that sold me http://home.lyse.net/brox/TonyPage4.html

I'll make a wild leap of assumption that you are writing about a 240 sedan of at least 1983 vintage. In the sedan, the backup lights are wired through the left side harness connector in daisy-chain style. The path to the right side is through the flex circuit - a wide spot of foil - and back out the harness connector. Check there. Black wires.
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Art Benstein near Baltimore

It's always darkest before dawn. So if you're going to steal your neighbor's newspaper, that's the time to do it.

How did you test that the bulb is good for backing up? Did you try put the good bulb from drivers side into passenger side socket? Look closely at the socket-loose wire? Bent contacts? Check the female socket for corroded contacts. if all testing fails, go to salvage yard for whole new back end to weld onto yours. (Sorry-I donno why I do that stuff-:oÞ~~~-hehe) Good luck.