Jon in Winchester, Ontario has a free standing natural gas stove and wants to get that heat distributed to the rest of the house.
First, what not to do
Do not try to put a grill in a return air duct of a forced air heating system over the stove as you see in the photo, especially forbidden for a wood burning stove. The return air duct is designed to handle room temperature or colder air. There are no heat clearances from the wood structure of the house as there are with the hot side of the ductwork. Potentially very hot air could rise off of this free standing stove and cause a fire somewhere else in the house.
Using Forced air ducts
If you want to use the fan system of a forced air system, you can use the return air ductwork as long as it draws air a good distance away from the stove itself, allowing the air to cool to a reasonable temperature before going into the return air duct. So even on the ceiling on the other side of the room would be acceptable.
Room to Room fans
If you want to move the air to an adjacent room, you will find that there exist through the wall "room-to-room" fans precisely for this purpose. You can usually find these fans at speciality wood heating stores. As for Jon's idea to cut holes in the ceiling above to let the air upstairs, that is exactly how Grandpa did it with his old pot belly stove. One detail that is necessary, Grandpa left the stairwell open to allow the rest of the circulation loop. Air that goes up has to find a path back down.
Air distribution works against sound proofing
Don't forget, that every effort you make to circulate the air from room to room, will reduce sound proofing and privacy between those rooms.