If you're talking about, say, something more along the lines of a natural gas fireplace/stove (which do function as heaters), from what I hear, generally speaking they're safe. (Don't misunderstand me; if they weren't safe, they wouldn't make and sell them. So, let me move on ...)
If you or anyone suffers from common headaches, nausea, or the like, ventless heating may not be the way to go. Also, the bigger the room/space you're heating, the better when it comes to ventless. Ventless natural gas heaters are very efficient, upwards of 99.9%. Thus, there's little 'by-product' expelled into the air. Don't let this fool you. Obviously, the higher the percentage, the better. But even with a 99.9% efficient unit, if the unit is operating over a prolonged period of time, by-products will slowly build up. Again, this is most prevelent in small rooms.
I guess the best advice I have for either ventless heates and/or a fireplaces/stoves, shop around, spend the $$$ for an efficient unit, and don't over-use the unit. Older homes are less-tight than new ones so they can be more forgiving. But new homes are tighter so ask a lot of questions and shop around to learn. My best to ya and hope this helps.