As a layman, I would like to ask the question in reverse. If a hole was drilled from the South Pole to the North Pole, would the stone stop in the middle. As we all know, stones dropped in a hole drilled at either location will fall, so my guess is that any stone dropped from either location will stop in the middle - where there is no gravity. Gravity at each location causes the stone to fall in, I am darn sure a stone dropped in a hole at the South Pole is not going to come shooting out at the North Pole. Stones dropped in these holes come under the laws of escape velocity.
The same applies to a hole drilled from one side of the equator to the other side.
