Instinctively to me, media side is better because if you have RF currents coming in, they're choked away from ground, whereas without, that flows right into whatever termination you have (HoJo resistor + ESD cap). You're relying more on the CMRR of the transformer, something the primary side CMC can't address.
It's less common for PoE, I think because CMC on pairs would trivially saturate; it's only meaningful for pair-pairs at a time, or the whole cable. So a more complex winding is needed. Maybe this isn't a hard limit after all -- good working examples of PoE apparently exist* -- but maybe it's harder to do than just tossing a power tap at the line and assuming everything else is normal.
*Mind, I haven't done anything with PoE yet, so my experience there is limited.
Also, especially with respect to GBASE's, there are full balanced vs. push-pull drivers (the latter needs VDDA to center tap), generally GBASE being the former type, so the transformer can be simpler, permitting higher bitrates.
Tim