Oh, ********. Everyone has an opinion here. That includes health and industry professionals as well. Here are a few points to consider from both sides.
1) Ventless gas heaters have been used for decades. They are not new. I'm still trying to find that national outcry due to "safety" since over decades of use, one would think there would be some data behind the hysteria.
2) Ventless gas heaters do not produce a gallon of water per gallon of gas. They can introduce 1 ounce of moisture per 1000BTU on average. That may or may not be significant depending on your particular application. Now, think about something else here. I don't know about the rest of you, but in the winter I'm running a HUMIDIFIER in my home. NOT a dehumidifer. And, I keep a pot of water on my wood stove to add humidity. So, adding that water vapor to pretty dry air is not necessarily a bad thing from where I'm sitting.
3) The whole "Just do it right" thing is simply somebody expressing their own opinion. I personally think nobody should drive a car, so for all of you out there thinking about painting your car yellow or buying a new yellow car, DON'T. Just do it right. There is no factual data to suggest that properly installing a ventless unit is not "doing it right". Sorry.
4) Ventless gas heaters are approximately 99.9% efficient. Vented in real life are maybe 80%. That's a big difference.
5) Most ventless units have O2 depletion sensors. Also they should be used with C02 monitors. If the 02 and C02 sensors say it's OK, you're not getting tons of extra humidity, and no smells to speak of, you are safe. Period. Correct that - if the sensors say it's OK you're safe. The other two variables really don't involve safety.
6) Consider your environment. How many sq/ft? Just how "tight" is the room really? Consider just how much C02 and 02 impact you're having BASED ON THE AMOUNT OF HEAT VS THE VOLUME OF THE NON-OBSTRUCTED AIR. Most likely you'll be surprised that the numbers aren't even measurable.
7) "Power-Vent" or "Direct-Vent" heaters are a better choice if you are having humidity issues for some reason. Or, perhaps if you have a respiratory condition that makes you far more vulnerable to any breathing related issues - just to be extra cautious. Or, perhaps if you live in a more temperate climate where the temps aren't dropping down to what we experience, and don't need quite the power or efficiency. But, if that is the case, I would also expect you to not be using a gas water heater, or a gas range in your home, as neither of them are truly vented either.
I can certainly see that especially in high humidity areas in a smaller volume room with tight seals everywhere, humidity could be a larger factor than what I experience, so YMMV. But I've got NO windows to crack to begin with. I've got a double wide garage door and two 8' wide garage doors - and one door leading to the house. None are opened except if a vehicle needs to come in or out - or somebody needs to walk in or out. Concrete block walls, front half of garage built into the hillside. House on top.