How can I repair this unusual joint that leaks?
The typical mid-trap joint of a plastic P-trap does not have a washer. Instead there is a tapered end that fits down into a smooth beveled hole on the opposite piece. The tightening of the nut forces them together to form a tight seal. Your's looks to be this type.

If this joint is leaking for you then there are several probable causes.


There is crud on one or the other side of the tapered part of the joint that keeps it from sealing 100% when the nut is tightened. 
The nut may have cracked and is not able to hold the necessary pressure to keep the tapered pieces together. 
One or the other of the pipe joint ends may have cracked due to aging or faulty plastic.
The smooth tapered parts on one or both parts of the joint may have been cut or scratched.


It may be possible to stop the leak by a thorough cleaning of the parts. If it is a damaged part then the simplest fix may be to purchase a new set of trap pipe, nuts and washers and just replace the thing. These parts are not at all expensive.