i think it should but im not sure.
If we assume that the Earth is perfectly spherical and uniform (which it isn't really), then we would experience no gravity at the centre of the earth.
I don't think we need to assume the earth to be spherical; for as long as it is smooth and symmetrical...you know, it could still be a bit flatten at the poles and wider at the equator...but for as long as it is symmetrical then gravity from one side cancels gravity from the diametrically opposed side, etc. and so, if you were at the center of an earth like this, you would experience no gravity from earth...but you will still be subjected to gravity from other bodies (moon, sun).
You've got to be right in the middle though.
An interesting, although not completely related case is that of people actually being on Earth. The atmosphere has mass, and it's not completely negligible...so the gravity just above the troposphere is slightly more than you'd expect. And all because you have to take into account more mass...
It must have symmetry but also must have symmetrical density
It doesn't need to be perfectly symmetrical however. The gravitational pull of one tonne, or 10^5 tonnes or any other relatively small amount (we're talking gravity here, the weakest of the forces) won't have a huge difference.
Oh yeah, in fact, it doesn't need to be regular at all and it doesn't need to have uniform density.
A slightly irregular shape can have an irregular distribution of density and still have C.O.M. in its "centre"
There was a pretty long and complicated thread a few months back about this. It boils down to this: At the center of the Earth you have no NET pull in any direction. The force of gravity of all that mass surrounding you cancels out in every direction. In effect you should feel like you are weightless. However, you are still under the influence of a gravitational field and would experience all the normal effects such as time dilation and such just like you do here on the surface, but at a slightly greater effect due to being in the middle of the mass instead of just on one side. Of course you would never survive being in the center of the Earth without being crushed...
Separate names with a comma.