Can I control two lamps from one pullchain?
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.