if you have a full kitchen sink and you hear a gurgling noise as the water is draining out this is quite normal and nothing to be alarmed about. The gurgling sound is a result of physics, when the sink is full the weight of the water above the drain pipe becomes quite airtight and as the water is being pulled down by gravity, the force of the gravity makes the water turbulent and brings in air with it, the gurgle sound is actually the air trying to escape up out of the pipe and into the sink, this is when you hear the bubbling or gurgling sound.
There is a very uncommon way to prevent this from happening and that is to install a non-return valve in the downstream piping at the bottom of your sink, but this is really unnecessary and it can also create more serious problems then just the gurgling sound, it's rarely ever installed because it's not necessary.
Check That There Are No Clogs in the System
There could be a clog in the sink drain or bathroom sink that could be causing air to build up in the system, for a clogged drain try to use a plunger or drain cleaner to get the clog out so that there is free water flow into the kitchen drain. if that doesn't work then you should attempt to clean the p-trap.
Do You Have an Incinerator?
If you have an incinerator and you're experiencing the gargling sound then you should contact your plumber to make sure that when you installed the incinerator that there is enough drain pipe from the garbage disposal to the p-trap, as sometimes with older houses the p-trap is not lower enough to work properly with the garbage disposals.
Does the Gargling Happen Even When You’re Not Using the Sink?
Another reason is that there could be insufficient venting or something that has blocked vent pipe or possibly also a partial blockage downstream. The flow of water on another floor could create some pressure imbalances, mostly in air but could also be water and this pressure exert themselves on the water that's stored in the p-traps below the sink. If it is happening very frequently then you should definitely get a professional plumber to assess vent system the situation because it might be releasing toxic gasses from the sewer line back into your kitchen which can be harmful so you should get someone to figure out what is causing the issue in the plumbing system.