I am not an expert, but by the sounds of it you have effectively got one side of the cooking area open to the outside, in which case there will be ample air supply to the stove and no carbon monoxide should be produced. If it is a gas stove then if the flame is blue and clean that is an indication that there is enough oxygen. Carbon monoxide forms either because there is limited air supply or because the burner is poorly set up. If your stove flame is yellow or sooty I would be a bit concerned. I think barbecues have been the cause of deaths in tents, they always produce carbon monoxide, because of the way they burn and continue to do so when they appear to have gone out. Barbecues should never be taken into tents. Never use a stove to warm up an enclosed area in a tent, there is the potential to consume oxygen faster than it can be replenished, leading to carbon monoxide being produced.
Chemeger - you seem like you might know the answer to this question. We had a new CO alarm for our new to us very old caravan for last weeks holiday. On nights we didn't open windows or long days out it went off. We didn't have the gas cable attached and it was checked at the ports too.