This is based on a 10 minute web search and some chemistry knowledge:
Glycerol decomposes into acrolein at ~535 F (~280 C). Acrolein is an aldehyde that is a decomposition product of glycerol. Glycerol is a basic component of a triglyceride. Basically fats in foods are composed of long chain fatty acids that are bound, in triplicate, to glycerol. Each glycerol can hold three fatty acids, which may be identical or may be different. As an ester, triglycerides can decompose into an alcohol and an acid (in this case, a free fatty acid) when heat is applied. There are probably other reactions that occur; maybe an organic chemist in the audience can enlighten us. The temperature at which this decomposition occurs is related to the type of fatty acids that are bound to the glycerol. Once the triglycerides decompose into free fatty acids and glycerol, the glycerol itself may decompose into acrolein if more heat is applied. You may have heard of a "smoke point" for cooking oils. Basically when you heat a cooking oil, you gradually cause the triglycerides in the oil to decompose into glycerol and free fatty acids. Further heating causes the glycerol (and the free fatty acids) to decompose (and combust, if you heat enough), which causes smoke and an acrid smell. The smell is caused by aldehydes like acrolein. This is also why once you smoke an oil, you should throw it out - those decomposition products are bitter, and frankly toxic, so you shouldn't eat them. Different oils smoke at different temperatures because the fatty acid content differs, and also oils that are more refined (think virgin olive oil versus purified oil) have less impurities that initialize decomposition reactions at lower temperatures.
Anyway, point is that pure glycerol is pretty stable until high (~500 F) temperature is reached. If glycerol decomposed into acrolein in appreciable quantities, you'd probably smell it. My guess is that the name acrolein derives in some way from the root word "acrid", because that's what it smells like.
From what I can tell, fog machines work (generally, there are different types) by introducing a proprietary smoke liquid, then heat is applied to vaporize the liquid. The liquid appears to be a mixture of glycerol and water (or some such). It's important to stress that you do not appear to be creating smoke, which is a suspension of solid particles in a gas. It seems what you are actually doing is creating a liquid-air aerosol. That is, you flash heat the water portion of the smoke liquid, which creates very fine droplets of glycerol that remain suspended in air. This has the appearance of smoke because fine particles scatter light. (This is very similar to what happens with dry ice, another popular method of creating "smoke" - carbon dioxide gas liberated from dry ice when it is put into water is very cold, which causes condensation of water in air to form small, suspended droplets, which scatter light in much the same way). Glycerol is probably used in this mixture, because it is relatively stable, nontoxic, and has (I guess) surface tension properties that allow it to form stable aerosols. The problem with dry ice fog is that once the fog is away from the vicinity of the dry ice, the fog warms, and the suspended water droplets re-evaporate and disappear. Thus dry ice fogs do not linger for very long. Dry ice is probably also more expensive than glycerol.
So, my GUESS is that the fog machine applies enough heat to flash-boil water in the smoke mixture, but not enough heat to decompose glycerol. If glycerol was being decomposed in appreciable quantities, you'd smell the aldehyde decomposition products pretty vividly. Therefore I would personally have little concern using a fog machine.
That said, it's possible that some acrolein or other decomposition products are formed at levels below the ability of the human nose to detect them. I have no knowledge of what the toxicity level of aerosolized acrolein is compared to its detection threshold, so I cannot offer any scientific guidance about whether fog machines pose a threat to your health, either chronic or acute, and you're probably not going to find anyone here who will give you a definitive yes or no on that score. Mostly because this information may simply not be available, and even if it were, the chemists here are not in the habit of making safety declarations for consumer products. That's a liability issue on top of the scientific uncertainty.
Hope you found any of that useful.
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