Why is one switch dependent on the other switch position in my double light switch?
I believe that you have mixed up the light-hot and the panel-hot wires. So when you flip the first switch it supplies power to the (whole) switch. You need to move the light wire (yellow in the picture below) and put it on the brass screw corresponding to the one you thought was the light wire. The wire that you thought went to the light is the power/panel hot.