Why would the kitchen sink be clogged on both sides?

If you have a double basin sink, more than likely each basin shares the same drain. You should check the trap to see if anything is caught in it (use a bucket to catch any water) Be careful if there has been any drain opening chemicals used recently (They cause bad chemical burns). If there is nothing in the trap, a drain auger would probably be the best bet. I would recommend asking around for the name a good plumber or roto rooter service man.