It's one of those unlikely "what if" things people (some with too much time on their hands) like to ponder. Like my question: What if you cored the earth from pole to pole, then jumped in the hole - what would happen?
There was a science fiction short story about that. I don't remember the name, but they used it as a prison. A perpetual pendulum of swinging prisoners...
I've thought of that too. I think you would be in a perpetual fall. As you passed the core, your rate of fall would begin to decrease as you would then be "falling up". You should come to a complete and brief stop just under the surface on the other side of the earth and then you'd begin to fall down again, reversing the process endlessly.
Assuming that all the hot stuff didn't come erupting out the hole? You'd fall back and forth for a long time before stopping near the core.
Probably about the same as a rat turd dropped in an hourglass.