Some runners choose not to wear a fuel belt or carry along a water bottle with them during a race. This could be for multiple reasons. Practically speaking, they just might not want to wear a fuel belt because they do not have one or find them uncomfortable. And carrying a water bottle, especially during longer distances like the half-marathon and full marathon, can become annoying and cumbersome. One of the main reasons most runners leave the water bottles at home during a race is because there are typically plenty of aid stations along the race route, which hand out water, sports drinks, and sometimes fuel depending on the race and how far it is. Not to mention, drinking TOO much during a run can pose an entire new set of problems, such as GI distress from having too much fluid in the bladder (which is definitely not ideal during a race!).
If you see runners just out on their morning run (aka not during a race) without water, chances are they were still smart about their pre-run hydration (and this certainly goes for runners during a race, as well). In fact, hydrating the body PRIOR to a run may arguably be more important than hydrating it DURING a run. Studies have shown that the best rule of thumb for gauging your hydration and for deciding on how much water is the right amount for your run is simply to drink when you are thirsty, and aim to get an average of 64 ounces of h20 down your gullet the day before your run.
How much water to drink both before and after a race totally depends on you and what your body needs. The best way to figure out what that is is to try out different amounts during your training runs. I would suggest starting with trying to get 64-72 ounces of water or sports drink within 24 hours of your run, then another 8 to 12 ounces before you leave the house just before your run. Then gauge how your body feels and adjust accordingly. And after the run, aim to put back what you lost through sweat (which is usually about 8oz for every 30 minutes) plus however much water it takes to take away your thirst.