Electrical Panel Grounding

All grounds must be bonded together at the service panel. This is for safety reasons, and to ensure that breakers will trip properly under fault.

Is the sub panel in the same building as the main panel or is it in an unattached garage?

What do you mean "with its own earth ground"? You cannot -- in the same building -- run a separate grounding conductor from a sub panel to a location other than where the house service is grounded. If there is a grounding conductor from a sub panel to, say, a water pipe that is more than 5 feet where the water pipe enters the house, the house water piping could be energized in the event of a fault. 

The sub panel must have a ground bar that is bonded to the sub panel and the circuit grounds get attached to the bar. That ground bar must be bonded to the main service panel ground at the point of entry by either a separate wire or metallic conduit to the sub panel from the main panel.

The sub panel must have an isolated neutral bus that prevents the sub panel circuit neutrals from being grounded upstream of the service panel, resulting in neutral currents raising the potential on the sub panel metallic parts and ground.

If the sub panel is in an unattached building and fed from a breaker in the main panel, then the sub panel needs its own earth ground (to a ground rod) and does not need a grounding conductor to the main panel.  

Earth "grounds" at different physical locations can have several volts of difference in potential. Depends on lots of factors. Only way to ensure it is at 0 Volts relative to AC is to tie them together. :)

Also, important, while you may have multiple grounds tied together, and even loop them, you may ONLY tie the neutral and ground together at one place, the service entrance. The service panel does not necessarily equal main. :)