You are right to point out that it depends on your definition of 'on Earth' - I implicitly took it literally to mean on Earth, rather than within the mass of Earth, but that was just lack of thought on my part.

Time at the centre of the core would, I suspect (but haven't done any maths on it), run slower than out in space.  The reason is that there is gravity there but it acts equally in all directions (more or less, let's keep it simple and ignore the non-uniformities of mass and shape!).  There is a gravitational field with no net effect because matter at the core is being pulled equally in all directions.  Time, however, is only affected by the amount of gravity, not the direction.

In truth, we still don't know what gravity actually is, or how it is 'made' .  For that matter, we don't fully understand time either.  I do know about digging ditches though.  The 10m x 1m trench I've just dug to lay electrical cables to some sheds has proven to me that I won't be digging to the Earth's core anytime soon...