Imagine two large metal plates having charge density $+\sigma$ and $-\sigma$ placed close to each other with a distance $d$ between them. The question is that why are the two plates are at different potentials. If a charge $+q$ is brought by an external force from infinity to the $+ve$ plate from the opposite side of the $-ve$ plate, the force experienced by the charge should be zero as the the $+ve$ and the $-ve$ plates exert an equal and opposite force of $q\sigma/2\epsilon_0$ on the charge. So the work done is zero for either of the plates. The work done or potential should not change when the charge is brought from any side of the plate as both are state functions. But Electric Field does exist in between the plates. And it should because one plate has $+ve$ charges while the other has $-ve$. So there must be a potential difference in between them.
I know that I have made two contradictory statements above but I can't understand what's wrong in my understanding. Please explain.