My question is about a home I added on to. The plans call for a 2 sided fireplace between living and dining room. The plans show 2 possible flues. One on each side. My question is asking what is the right way to flue a 2 sided fireplace? I am in the building process and the fireplace is finished except for some cosmetic finishing. After a rainy day we noticed water dripping down into the fire box. I looked up and it was a straight shot up to the top of the fireplace. Is this correct? Or do I need something redone? The damper is a large flat piece of metal and when I pull the chain it goes straight up and down. We did try lighting something to see if it draws (considering it has 2 sides thus 2 sides to draw from) the smoke came into the room rather quickly. What is my outcome going to be or what is it that I need to do to fix?

You can add a rain cap to the chimney to stop the rain but I'd install a closeable flue cap. That will safe heat and keep out the rain. Starting a draft does not always happen on its own. Slightly open a window or door, roll up a newspaper like a torch. Light the torch and shove it up inside the fire box as far as you can, this will start the draft.