Why does this sink gurgle?
I think the most likely cause is that waste water from the upper sink is being forced up into the vent pipe for the lower sink. 

The water descends without much restriction for about nine feet and accumulates enough kinetic energy to escape its intended path through the junction of the two drains. Some of it splashes into the lower drain and some into the vent pipe. This pushes air in through the lower sink p-trap, and then, when the flood abates and the water flows back out of the lower drain and vent, it pulls air out through the lower sink p-trap. 

The root cause of all this is that the join between the two sink drains is much too close to the lower sink. See the left half of the diagram below:



You can eliminate the problem by repositioning the join between the drains to a lower location. See the right half of the diagram for one way to do this. 

If you decide not to change the drains because it's a lot of trouble just to get rid of an annoying gurgling, you have my sympathy. Just run some water in the lower sink after each time it gurgles, to make sure the p-trap is filled.

However -- if you are smelling sewer gas in the lower apartment, then you must call in a professional plumber to change the drains.