Your trailer connection carries 12V positive to each light as needed AND a negative via a white wire to a screw which grounds (bonds?) the trailer frame.
The trailer frame becomes the carrier for 12V negative and is tapped in several places to accommodate each light.
Side lights typically have a 12V positive wire and another, negative wire that is grounded to the trailer frame.
If the lights don't shine then find the ground wire and ground it to some bare metal.
It that makes them light then sand or file the place where the ground wire goes and reattach with a new screw.
You can also test for 12V at the side light by probing the 12V wire and a bare metal part of the trailer.
To do this you will penetrate the wire and make it more likely to fail in the furure.
Good luck!
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