Modern wood stoves are dramatically more efficient than the typical fireplace you'd remember from youth.
If anything, the issue would be high radiant heating on indoor surfaces causing evaporation and out-gassing. I have heard people complain about this, but it would generally be more of a freshly-installed or newly-built issue. Or perhaps if the home in question is built for passive heating, meaning that it's practically hermetically sealed and hence would require active ventilation.
As for air quality, a rocket mass heater will get hot enough to put out zero CO, and most or all of the particulate will be broken down as well, but sadly there's only one device I know of that is UL-certified. They are very much a niche concept; most stoves and masonry heaters will be putting out some, and in a city there'll be enough that it'll have an effect on health.
Where I live, in the San Francisco Bay Area, the air quality regulator says that wood smoke is the main source of air pollution. It's not ships or trains or trucks or cars or refineries. It's fireplaces and pizza ovens.
My understanding is it causes the air quality to go down and some people are affected. I like the smell of burning wood, but I can see how it can lead to an ashy experience.