Will Sharks Attack Me If I Dive on My Period?:
Thankfully, sharks are not going to smell your blood and come chasing after you if you dive while menstruating. Studies have been conducted to observe sharks' attraction to human blood. Sharks appear curious, but not aggressive when human blood is in the water. In fact, sharks are most attracted to fish gastric juices (not even fish blood) which makes sense as a fish that is leaking gastric juices is definitely disabled and easy to attack.
Furthermore, a menstruating female loses only a few milliliters of blood a day. The majority of fluid loss due to menstruation is water and uterine lining cells. Most females will find that their period actually stops when they are submerged in water; the vaginal opening stays closed and the increase in ambient pressure helps to keep fluids from leaking out.
Diving While Menstruating May Increase the Risk of Decompression Sickness:
Diving on your period is relatively safe. However, studies have shown that scuba diving while menstruating may increase a diver's risk of decompression sickness. One study observed that females were almost twice as likely to experience decompression sickness during the first week of their menstrual cycle (during menstruation). Additionally, divers who were taking oral contraceptives (the birth control pill) were more likely to get decompression sickness than those who were not. This study showed a correlation between menstruation and decompression sickness, but more research is needed before conclusions can be drawn.
The reasons that menstruating divers appear to be more susceptible to decompression sickness is not understood. Suffice to say that bodily changes occur during menstruation that appear to make nitrogen elimination less efficient. Consider also that menstruation can lead to dehydration, which is a known contributing factor in decompression sickness.
As a dive professional, I dive every day of the month. I have yet to experience any problems due to menstruation. However, divers would be well-advised to dive more conservatively while menstruating. This includes making fewer, shorter, and shallower dives with ample safety stops than they would during other times of the month.
Diving with Extreme Premenstrual Syndrome/ Physical Discomfort:
A particularly condescending journalist wrote, "Cognitive changes occur with various stages of the menstrual cycle, and in theory a woman's ability to make safe decisions during a scuba dive could be affected by her menstrual state."  This statement makes me want to punch the author in the face, and I am not even on my period. What does he think I am going to do? Refuse to share air with my boyfriend because he told me I looked fat on the surface?
However, the author may have a point, even if it is poorly stated. Some woman experience strange side effects during PMS and menstruation -- physical dis-coordination, forgetting things, etc. Other women experience extreme physical discomfort. Getting all the way to a dive site and realizing that you have forgotten your mask, or dropping a weight belt on your foot is not fun. Diving with extreme cramps is just horrible. Consider that physical pain is your body's way of warning you that everything is not 100% okay. Be cautious or don't dive is you experience extreme PMS or side effects during your period.
Now we get to the nitty gritty, icky part of the the article. How does a menstruating diver deal with fluid loss on a dive boat? Underwater, most divers stop menstruating. The vaginal opening collapses, and no water or body fluids enter or exit a diver's body. Additionally, most divers use wetsuits, which limit water circulation. Any leaking fluids are likely to stay inside the diver's suit. You will not be diving in a little red cloud.
However, a diver on her period may need to control blood and fluid loss on the surface before and after a dive. Tampons work very well for fluid control, and can be left in during a scuba dive. In fact, because the vaginal opening usually seals shut during a dive, the tampon is unlikely to even get wet underwater. The same cannot be said for the tampon string, and this is when embarrassing situations can happen. A wet tampon sting can wick fluids down and out of a diver's body after a dive, and this can cause some leakage. My advice? Carry extra tampons and switch them out as quickly as possible after a dive, even between dives if a bathroom is available on the dive boat. Leave your wetsuit on until you are able to switch out the tampon.
The Take-Home Message About Diving During Your Period:
Most females divers (and all the female diver professionals that I know) dive during their periods. Some studies suggest that scuba diving while menstruating may increase a diver's chance of decompression sickness, so be sure to dive conservatively and stay hydrated when diving on your period. Divers who experience severe PMS or menstrual pain may want to refrain from diving until these symptoms pass. Finally, plan for logistical considerations, such as carrying extra tampons, ahead of time in order to avoid post-dive fluid leakage.
 "Women and Scuba Diving" JE Cresswell, M st Leger Dowse, 28 March 1991, PubMedCentralCanada.
 Diver's Alert Network (DAN)
 The London Diving Center Online, "Considerations for Women and Diving"
 Journal of Aviation Space and Environmental Medicine; 1992 July; 63(7) 61-68
 Journal of Aviation Space and Environmental Medicine; 1990 July; 61(7) 657-9
 J. Obstet Gynaecol; 2006 April; 26(7) 216-21 PubMed
 Journal of Aviation Space and Environmental Medicine. 2003 November; 74(11) 1177-82