What you describe is not uncommon. I'm no expert, but I believe it's caused by the friction from your running top rubbing against the nipple. I'm a male and I have the same problem. I've solved it by wearing "Nipguards," a product widely available in North America. You can find it and order it online. I wear them for long runs and they work fine. The only problem is that, sometimes, they fall off. It doesn't happen often, and it has hardly happened to me ever since I made a conscious effort to keep my running shirt as dry as possible. (I used to pour water over my head during long runs/races; I have stopped doing that.) I'm sure there are other solutions (some people just use vaseline, others use a bandaid or something). It is definitely a solveable problem.
Here is the official Niguards site:
http://www.nipguards.com/ (I'm younger than the man pictured, but I have one or two pictures like that!)