Never been bit by a running saw, but they can still git ya when you think they're sleeping......... sharpening w/o gloves, swinging climbing saw on tether, reaching in the dark and catchin' chain.. yup. Been there, bought the T-shirt.
I've had several close calls over the years, particularly in the first couple working professionally. Cut into chaps a few times in thigh area, once or twice in side of calf area. A few nicks in the tip of the sole of my right boot. Most of this stuff seems to happen at the end of the day or when fatiqued or working like a madman to get something done..... I've had the usual handsaw, polesaw bites, including breaking my right big toe while using 6 sections of Jamesons(that hurt like he!!).
Had several friends/co-workers hurt over the years. One of the worst was a friend that received a viscous laceration from lower neck to below the sternum from a kick back. They were felling trees on family property when one tree hung up. He hung the saw up in a topside cut(tree at 45*) and attempted to free it by doing a slashing cut with a second saw. Must've caught the tip, kicked back into his chest.