It may shock you, but get this: There has never been a widespread study conducted on the sanitation or the necessity of the courtesy flush. Before we nod knowingly about how it's just like Big Toilet to cover up the facts, you should know that we do have some ancillary information that might just give us a glimpse into the science of the courtesy flush.
In 2012, for instance, a study found that flushing the toilet with the lid up can cause a bathroom space to be sprayed with Clostridium difficile bacteria (also known as nasty little germs that cause serious diarrhea and inflammation of the colon) [source: Moisse]. Ergo, one can assume that unless you are standing up, closing the lid, flushing and sitting back down for more -- well, you might be coating your hindquarters with a fine mist of bacteria. Frankly, it doesn't necessarily mean you're unsanitary. If you're staying clean and hygienic (and your immune system is healthy), you're probably not going to suffer from C. difficile.
So you're not doing yourself any favors to flush the fecal matter occupying a bowl you're currently sitting on, but you're probably not causing grievous bodily harm either. But by doing it, are you even addressing the odor problem in the first place?
This one's a little trickier to figure out. If you time it right, you're making sure the poop is in the bowl the absolute shortest amount of time, minimizing odor. And this certainly stands to reason: The less time the fecal matter is exposed to air, the less time the sulfur compounds (produced by bacteria in poop) can work their smelly magic -- and note that water doesn't "cover" the aroma of feces, which anyone who has ever once happened upon an unflushed toilet or portable toilet will tell you. Liquids are perfectly capable of transmitting smells to lucky local folks, thanks to escaping gases.
So courtesy flushes may be polite, and even moderately effective, but they're a huge waste of water, as well (unless you wind up preventing a clogged toilet from overflowing). A newer high efficiency toilet uses around 1.3 to 1.6 gallons (4.9 to 6.1 liters) of water per flush; keep in mind that older toilets could use 6 gallons (22.7 liters) [source: EPA].
If you're in the habit of flushing more than once, you might get a nod of appreciation from your stall neighbor at the sink. But you won't be doing the environment any favors.