The name "common mode" means that the current on the shield and the center conductor is traveling in a common direction. This condition is a violation of the assumptions required to make a transmission line work.
For the transmission line to work correctly, the current must be in opposite directions on the shield and center conductor; in this case, the current will be on the inside of the shield and the outside of the center conductor, the induced magnetic fields will oppose and cancel, and there will be no leakage.
If there is common mode current, then these conditions are violated, and there will be RF leakage both into and out of the transmission line.
The world generally not being perfect, it is possible for both conditions to exist simultaneously. In other words, you can have both common mode current and opposing current on the shield at the same time. When this occurs, then presumably the common mode current will be on the outside of the shield and the opposing current on the inside of the shield and the portion of the current that is common mode will cause leakage. However, it would also be possible (but unlikely--so maybe only a small fraction of the current) for the common mode current to travel in other parts of the shield and cause eddy currents, which would heat the shield rather than leak. Poor quality coax might have a thin shield, which be less than the skin depth, allowing the two currents to mix and allowing eddy currents to be worse.
Typically, common mode current in transmission line is caused by the shield of the coax coupling with the antenna, so that matched opposing currents flow in the antenna and the outside of the shield in addition to the matching currents on the inside of the shield and the outside of the center conductor. This coupling essentially makes the shield part of the antenna.