You can do whatever you want... but to be safe, no you should not share the flues. It is also against firecodes.
Wood creates creosote and the water from your gas unit will destroy the pipe. Not only that but you can get backdrafting and carbon monoxide into the home.
I would send the wood, double walled pipe through the roof
Your local firemarshall can help with questions also, and inspect the installation so your insurance will cover you in a fire.