I have two bedside swag lamps, one on each side of the bed. Each one has a plug that goes into the wall. They each have a pull chain for use in turning them on. I would like it if, when I pull the chain on one lamp, they both would turn on. Is this even possible and if so how would I accomplish it? I am hoping there is some special plug I can buy, plug both lamps into it and be done...
A simple solution would be to use a surge protector outlet with master control. This would be a power strip that has one outlet labelled master, some others labeled master controlled, and some that are always on. The master outlet is monitored for when that device is consuming power, and the master controlled ones will only be on when the master device is on. These are typically used with computers, where the tower is plugged in as master, and peripherals as controlled, so that for example, speakers and monitors are only on when the computer is on as well.
So in your case, you would plug one lamp into the master outlet, and the other into one of the master controlled ones. Since your lamps are next to the bed, the power strip can be in the middle behind / underneath the bed.
Here is an example of what one of these looks like (not endorsing a particular brand, there are many different ones available). In this case, plug one lamp into the solid green, and one into the green outlines outlet. When the lamp plugged into solid green is on, the other one will turn on as well.
I've never seen a wired set up where when one light turned it, it also turns on another device. You can, however, go with wireless devices. They range from basic RF (radio frequency) devices to z-wave and WiFi devices.
If you go with a smart device, you can add other lights, locks, speakers, etc., over time. Google Home, Samsung Smart Things and Amazon Alexa are examples of smart device hubs. You need a hub if you go this route and make sure the devices you buy work with the hub.
Here is a simple RF device just as a example (not a smart device). You'll need to research it and other like items to see what would work best. https://www.amazon.com/dp/B01HWB0TPU/ref=cm_sw_r_cp_apa_C.MYBbT3DPGYV