I don't get particularly red but I sweat a lot!!
I have recently joined a running club, and went out for my second run with them yesterday. It was a fairly difficult hill-repeat session. When we got back to the clubhouse, we did a few stretches and one included sitting on the floor to stretch the legs. When I got up, there was a massive damp patch from where I had been sitting!!
It was a hard session, but it was only 5 miles in total. After only a mile gentle jog, I feel myself sweating and my back getting wet and starting to stick to my running shirt. I often wonder if it's something to worry about, but I don't think it is....it's just me 
I don't know if it's my general size/build (6ft 3, 13st 10) that causes the issue...?