Interior chimney pipe is designed to radiate heat to the inside of the room. It is thin walled, typically 18- or 24-inch sections. The exterior pipe is insulated, double or triple walled and designed to retain the heat to protect roofing and siding. They both separate the same way. Over time, creosote, a thick black gunk, builds up in your chimney pipe and you can't remove it unless you separate the pipe to get access to the pipe interior. It's easier than it looks.
Loosen and remove two pairs of small screws, one at the top and one at the bottom, of one section of pipe using a small screwdriver. It doesn't matter which section you choose.
Put on some gloves and grab the pipe with both hands. Twist it back and forth. If it will not twist, bump it side to side with your palms until you hear a gritty sound, which means you have broken the creosote from the inside. Twist it back and forth again until it moves slightly.
Lift up as you twist the pipe. Bump it side to side, twist and pull up at the same time. The two sections overlap each other typically by only 1/2 inch.
Break one side loose by tipping it sideways and then pull down on the pipe to free it from the one above it. Wiggle and pull the pipe side to side until the top and bottom ends have come loose.
Hold the section diagonally and pull it free from the other two sections. Remove successive screws and pull straight down or straight up to remove each section, one at a time.