Currents in grounding wire
Ground wires should not carry current except during faults. If the ground wire carries any current there will be a potential difference between different grounding points (bacause the current flowing in wire causes voltage drop because wire resistance). This is why a common wire which works as neutral and grounding wire is very bad thing.
When there is separate wiring for grounding you can't still completely avoid the current flowing in grounding wires ! There will always be some capacitive leakage current form the live wire to the ground wire. This capacitive leakage current is caused by the fact that the wiring, transformers and interference filters all have some capacitance between the ground and live wire. The amount of current is limited to be quite low (limited to be between 0.6 mA to 10 mA depending on equipment type) so it does not cause dangers and big problems. Because of this leakege current there is always some current flowing in the ground wire and the ground potentials of different electrical power outlets are never equal.
The leakege current can also cause other type of problems. In some situations there are ground fault detect interrupter (GFCI) circuits in use the leakage current caused by many equipments together can make the GFCI to cut the current. Typically GFCI circuits are designed to cut current when there is 30 mA or more difference in currents flowing in live and neutral wires (the difference of those current must flow to ground). Some GFCI circuit can cut the mains feed even at 15 mA leakage current which may mean that if you connect many computer equipments (each of them having 0.5 to 2 mA of leakage) to GFCI protected power outlet you can cause the GFCI to cut the power feed.