I went swimming yesterday. I would not have done so without a tampon in place. If it's light flow it's probably possible, but not on the first couple of days (for me). But I've also spent my time as a lifeguard, and I wouldn't choose to have anyone bleeding into the water (nosebleed and banged head are probably more likely, though.)
Period blood is not much of an issue compared with diarrhoea down one's legs - I had been blissfully unaware that people would run through that. Periods at least are normal - diarrhoea isn't, even if it is common for marathon runners.
I think there are still some taboos around periods in this country. I can say, "just got to go to the loo," at work, even while holding a spare tampon. I think if I said, "I really need to change my tampon before I leak," most of my male colleagues would probably be shocked, even though nearly all are married with children, so should have some idea about menstruation. Some of that is normal, in that I don't need to know he needs a poo desperately, and in any case, we tend not to announce we're going to the loo, except in the context of, "can you spare me some time to go over this report now?" "yes, but only after I've been to the loo."