When I was a rookie cyclist someone called me out for having chain grease on the inner part of my calf. Said they were “Cat 5 marks”. Well, then! Being such a newbie I had no idea what that even meant, but it was taken as a loving insult. 
Cat 5 is short for "Category 5", the slowest and often least experienced class of riders in competitive cycling. By tattooing myself with chain grease, I branded myself an inexperienced rider who repeatedly made contact with the chain or chain ring on the right side of the bike. Experienced riders know better than to mount/dismount on that side, you see.
Recently, I wrapped up a group run with Fairmount Park Conservancy (join us!!) and noticed a friend had dirty scuff marks all over the inner aspect of her calf. She looked at me and noticed I didn't have any. Was this just because I am awesome? Ha, hardly ;-) But ok, yes.
The scuff marks are a result of your swing leg foot swiping the stance leg calf as it passes by. I'm sure that much is obvious, so here's a little play by play describing what's causing that to happen. Try to follow along:
Your right foot hits the ground.
Your bodyweight transfers to the right foot.
Your body doesn't quite react and stabilize in the lateral plane well.
So your pelvis drifts a little farther to the right than it should.
Meanwhile, your left leg is swinging through.
Since the pelvis has drifted too far to the right, your right calf is now “in the path” of your swinging left foot.
SCRAPE! Your inner right calf now has scuff marks on it. :-(
Can we fix this?
Yes! Here are the steps I'd take.
First: if possible, get assessed by someone skilled in Neurokinetic Therapy. We can figure out what's compensating for what. Often there's a motor control issue with the lateral “sub-system” where one or two muscles of the three aren't functioning properly.
Do you have a previous history of severe ankle sprain? This could still be causing issues with your motor control. Even long after the injury has healed, the brain could still want to 'protect' the ankle by inhibiting key hip or core muscles.
In the gym, practice the suitcase carry and walk with a few inches width between your feet. Do NOT let your feet cross paths! This exercise will really let you know where your “side abs” (quadratus lumborum and obliques) are in a hurry!
While you're running, practice running along a crack in the sidewalk.
Only allow one foot to touch the crack. If you have trouble stabilizing in the lateral plane, you may find this quite challenging. Start slow!
So, a few factors to consider, but the first one is motor control. Get assessed if possible, but if not, be sure to start slowly. Gotta be able to control the move slowly before you can expect to do it well quickly.
**CAVEAT** This phenomenon can also be due to having a toe-in posture. It would then be important to determine if this is a fixed anatomical issue or a compensation for a lack of core/hip strength and motor control. In my experience, most people fall into the camp described above, however.
What about you? Ever wonder why the heck those dirt marks are all over one (or both) calves after a run?