Wood stoves, like all fuel-burning furnaces, need to take fresh air in to keep the fuel burning at the optimum rate, and also need to expel the smoke and harmful emissions safely away from the home interior. Vents at the bottom of the stove allow fresh air in, and a pipe, also knows as a flue, takes the smoke away and through a wall or roof. A scrubber in the stove cleans the smoke of most harmful chemicals, but not all.
Creosote is the primary problem with burning wood as a heating fuel. This tarlike substance is left over as residue from the chemicals in the smoke as it comes into contact with a flue wall that is slightly cooler than the smoke. Creosote is flammable, and if the buildup becomes significant, it will ignite. A flue that does double duty servicing two stoves will get twice as dirty and the risk of fire greatly increases.
The diameter of a properly installed flue is measured as a percentage of the door opening of the stove, a number that is provided in the installation directions. If an existing flue suddenly gains another wood stove that feeds smoke into it, the size of the flue will be much too small to properly vent it. This increases the risk of smoke backing up into the room when one or more of the doors is open.
Contrary to the way most people view smoke, it does not flow upwards. It flows from the area of greatest pressure to that of the lowest pressure. If the smoke from one stove is flowing through the vent pipe and the lowest pressure is not upward, but rather in the direction of the other unutilized stove, the smoke and carbon monoxide will flow to and out of the other stove. This is taking a chance that the home will fill up with smoke that creates a nuisance at the least, or harmful fumes at the worst.