If an operator is also wearing a smock, but is not electrically connecting the smock to either their body's skin or ground, then the smock is providing only partial protection. Charges on the smock may have no where to go or discharge to if the smock is not grounded. A popular way to ground the smock is with a coil cord either attached to a snap on the waist area of the smock or via a wrist strap snapped to the inside cuff of an ESD smock. If the mat is grounded at several locations, then there is the probability for ground currents to flow, which will give false readings to a meter movement when measuring Ohms. Other possible causes would be a low battery in the meter, ground loops between the meter's AC power and the ground it is measuring.
Some digital meters are sensitive to magnetic or strong electric fields in the near vicinity.
Check for power distribution cables, EMI sources, etc. Try repeating your measurements in a shielded area to see if your problem is eliminated. KEY Not necessarily, but you are right that when current passes through a 1 Megohm resistor that there is a potential difference between either end of the resistor. The key in ESD control is to eliminate the charge on the body or other conductor within a safe discharge time. If the discharge time is 10 milliseconds, then there is a potential difference between the ends of a 1 Megohm resistor from 0 seconds to 0.01 seconds.