Nestor Kelebay-Thanks for the info. I have s-traps on my kitchen and bathroom sinks. Some water does stay in the trap after using the sink, although it doesn't look like very much. No sewer smell. Anything else I can do/check to stop the gurgling and echoing I get after using the sink?

 

Your problem sounds like a lack of air vents.
When water drains it must be able to pull air behind it. If it is starved for an air source it will drain slowly, glug and pop, and pull water out of the trap seal. An example of the need for venting is to take a milk jug fill it with water and turn it upside down, it will drain but only as air is forced in from the opening (glug) now cut a hole in the bottom of the jug and you will see a big difference in how it works, drains work on the same principle.
To quiet these drains you will need to have air vents installed, which I'm sure the landlord will love to hear.The proper way to do this is to run a pipe through the roof.
The cheap fix, if its code approved in your area is to redo the traps and have automatic air vents installed under the sinks.
I'm not a fan of these but it will improve your drain situation alot.
Best of luck-plumber john