We’ve had plumbers snake our sewer drains before as tree roots have infiltrated the line from our house, but typically when our kitchen and bath sinks clog up we break out the plunger and gently try the clear the line. In a two-sink kitchen make sure you’ve covered the opposite drain opening with a rag or something to create a seal or else you will simply blow air into the open drain and not through to the clog in the line.
If your plunger doesn’t do the trick then consider checking the p-trap located directly below the sink and if it has a drain plug remove that and pull any excess junk that might be in there through that plug. If there is no plug then you can remove the p-trap as debris have have collected at the bottom of the trap or even at the curve in the top.
If you have pulled the p-trap out you may want to consider snaking the line for good measure using a drain auger. Make sure to go easy on the pipes and remove any excess debris. This should ensure a clean working plumbing system below your kitchen sink.