If those vents are truly clear, I don't know how this would be happening. However, since you say your roof vents are clear, I suspect that particular sink drain not is not properly connected to those vents or its passage to them has become blocked. I'd begin by plunging and/or sealing and blowing down with compressed air. If there is a localized blockage to an otherwise proper connection, that may open it. This is easy to do. I'd blow both sink and shower drains. Do it with the drains open first to check flow-through to the other. Then seal each and blow separately. Suspect you may learn something by just doing that. If it doesn't solve the problem, you'll have to dig deeper.

Do you get sewer-gas smell or bubbles from either drain when they're idle or only when they're in use?

Thanks for the reply, Asolo. There isn't any smell from those drains when the other one isn't in use. I'll try the compressed air thing, and see what happens.

Also, are your sure both of these drains were installed with traps? Only other thing I can think of.