When a wire is carrying current, charges enter the wire at the positive terminal of the voltage source and exit at the negative terminal, keeping the overall charge at zero throughout the current's passage.
Charge moving across a conductor produces current. There is always the same number of electrons in a wire today. As a result of an equal amount of electrons entering and exiting the circuit simultaneously, current flows through the wire when electrons enter the circuit. Current flow may nevertheless be neutral with no overall charge.
Neutral wires do indeed carry current. In single-phase circuits, the current flows in the opposite direction from the "phase" or "line" or "hot" wire, but it is exactly equal to that current. Out of "line," into "neutral," back into "line," and finally back into "neutral" and back into "line" (since it is AC the direction reverses every cycle.)
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