Are all Earth Grounds Equal?
Are these equivalent circuits?


Not really.  The resistance between G and N on CCT#1 is likely to be very low.  It may be less than an ohm.

The resistance between G and N on CCT#2 is likely to be much higher.  With a simple rod hammered into the ground, it could be anything from a few tens of ohms to a few kilohms.  Dirt isn't a good conductor, especially if it's dry.

The first supply would be classified as "TN-S", the second is "TT".


  What problems could occur when a circuit is grounded elsewhere than where its electrical service is grounded?


You haven't said what circuit protection devices CCT#1 and CCT#2 have.  A short between L and G on CCT#1 will cause a large current to flow.  This should trip a circuit breaker or blow a fuse.

A short between L and G on CCT#2 is unlikely to pass enough current to trip a breaker or blow a fuse, unless you have an unusually good earth rod.  Such circuits really need to be protected by an RCD/GFCI to be safe (RCD and GFCI are the same thing under different names).