A double kitchen sink that is draining slowly could be clogged with debris such as food particles. A snake is a tool that can be found at a hardware store that is used to remove things clogging flow in a drain.
Partial clog or not enough or proper venting. If it is a single sink, there may not be a vent to it. On a double sink, one side vents from the other. On a single the water is trying to push past the air in the drain.
The sink doesn't have an adequate vent.
The trap and drain in most sinks gets it's venting from the overflow tube in the sink or in the case of a double kitchen sink, one side gets it from the other. One problem is with most cultured marble and cast sinks. Most of these do not have an overflow feature and may drain slow.
Kitchen sink drains are a fertile environment for certain forms of life, and these can cause the drain to run slow. When hot water is run down the drain, it kills some of the lifeforms, allowing the drain to run freely for a while. Try using a drain cleaner, or a drain snake, to clean out the drain. Never pour grease down a drain, as it cools and solidifies quickly in the pipes.
One way to open a slow kitchen drain is by using a plunger on the drain. If the drain is still slow put some baking soda and vinegar in the drain to loosen any debris that remain in the drain.
1- Increase the diameter 2- water jet the lines to scour them back to full flow as designed
Because both sides of your sink are connected to a common drain line, and the common drain line is slow.
BAD Venting
The vent pipe could be clogged. The vent pipe is that pipe that pokes out of the roof, there will probably be several. You may be able to clear the blockage by running a garden hose down the pipe (start by picking the pipe nearest the sink area) and turning it on to full blast.
If you have a stink coming up from the overflow drain in your bathroom sink, you have some garbage down there that you should not have. You probably have a lot of hair clogging the drain. It is probably slow draining. You need to clean it out and get rid of the gunk.
Call a plumber to snake out your kitchen drain line. Do NOT pour acid down your drain. Do it right and it will be almost good as new. (I'm an optimist)
Pour 1/2 cup of baking soda down the drain followed slowly by white vinegar. It will foam and bubble. Let it sit about 5-10 minutes, then flush the drain with hot water. It will also remove odors from the drain. This works best in a kitchen drain that is running slow, not completely stopped up. I don't think it works as well on hair clogs.