I'm late to the party, and the other doctors have done a nice job of explaining the issues. Nevertheless, as a hematologist (blood and clotting specialist), let me take a stab at clarifying a couple of points.
First, it is correct that there is no "blood thinning" going on. Whichever doctor first used that term decades ago to talk to patients should be shot. So-called "blood thinners" such as warfarin (Coumadin) interfere with vitamin K metabolism which, in turn, inhibits the production of certain clotting factors thereby interfering with the body's ability to form a clot. The blood does not get thinner, but it might seem that way because bleeding will be very slow to stop.
Aspirin and other drugs which have an aspirin-like effect prevent or reduce platelet aggregation. Platelet aggregation -- or clumping -- is often the first step in producing a clot in an artery or vein. What happens is that the blood vessel wall gets damaged (by trauma or by cholesterol deposits or maybe by nitrogen bubbles -- the last one is conjecture). The damaged cells release substances which activate platelets that happen to be floating by. Those platelets release more substances which cause the platelets to clump in order to plug the hole or to cover up the damaged area. When the platelets clump, they release other substances that activate clotting factors which are proteins that also float around in the blood. This is great if you have a hole in the vessel, but not so great if the blood vessel doesn't have a hole and supplies your heart muscle or brain.
In any case, you can see that there are two ways to try to stop a clot from forming. One is to stop the platelets from clumping, and the other is to prevent the clotting factors from building a full blown clot. Aspirin, Plavix and similar drugs stop the platelets, and drugs such as warfarin and heparin stop the clotting system.
As for the dose of aspirin, it has been found that a single 81 mg dose of aspirin is all that is needed to poison platelets for a whole week. This is why you are told to stop taking aspirin and NSAID's a week before surgery. When you want to stop platelets, though, you have to take aspirin daily because you are always making new platelets. If you took aspirin yesterday, those platelets are inactivated for a week. But you are making more platelets today, so you have to take more aspirin if you want to inactivate those, too.
As far as we know so far, the most important factor affecting DCS risk is blood circulation. Although one can "thin" the blood a little by hydrating, the effect is minimal because your kidneys cause you to pee out any extra fluid that might tend to thin your blood more and more.
It may be easier to understand if you think about what happens if you are dehydrated. The body contains miles of blood vessels. Much of this is "reserve circulation" through which blood has low flow. The fluid in these vessels can be mobilized when needed to maintain proper blood pressure or to help control body temperature. For example, when you get hot, your skin turns red because the body opens up vessels in the skin to try to dump heat. Conversely, when you get cold, your ears and hands may turn red as your body tries to increase circulation to transfer heat from your core to those exposed areas.
Well, this mechanism depends upon having enough fluid to be able to shift around. When you are dehydrated, your body may have a harder time shifting fluid around where it needs to be.
How does this relate to DCS? Smarter people may know for sure, but my understanding is that the body needs adequate flow to all tissues that are trying to get rid of accumulated nitrogen. Some tissues have more blood flow than others. Joints, for example, have little blood flow. The cartilages get their nourishment and lubrication through fluid contained in the joints. The brain and spinal cord get nourishment and help through blood, but also from spinal fluid that bathes them.
If one is even partially dehydrated, then the flow to and from these tissues can be compromised to the point that nitrogen coming out of solution is not carried away quickly enough, but instead the emerging nitrogen microbubbles collect and form larger bubbles which, in turn, can start causing symptoms.
Again, we do not really know if this is what happens, but we do know that adequate hydration seems to help lower the risk of DCS. And if you accept this model, then you can also see why aspirin and other drugs may not really lower the risk of DCS that much. They may help only insofar as they help maintain adequate circulation in those individuals who already have compromised blood flow from aging or diabetes or high cholesterol levels or smoking.
One possibility is that aspirin and other drugs, like ibuprofen, may only appear to lower the risk of DCS by masking the symptoms. That is, they may not actually prevent DCS, but they may keep one from feeling the pain because they also have a pain-lowering effect.
As debersole pointed out, the risk of DCS is quite low in the first place -- maybe 2-3 cases out of 10,000 dives. This does not mean that one need not worry, because it is still real -- very real if you are one of the 2 or 3. But it does mean that the incidence is so low that if one wanted to perform a study to find out whether aspirin (or any other agent) helped reduce the risk, you'd have to track millions of dives in thousands of divers over several years. Logistically, this would be impossible because what are the odds the non-aspirin group might take some aspirin accidentally or without knowing it? There would be no way to control the group, even if you used pressure chambers to simulate dives.
Hope that helps more than it muddies the water...