The NEC deals with antenna systems in article 810. It specifically states the need to connect (bond) any added ground rod to the existing electrical service grounding system, using #6 AWG copper and approved connectors at a minimum. This will ensure that there will be very little difference in voltage levels between the new ground rod and the rest of the grounding system on the premise.
So the best practice is to use only one grounding system. (Which can include a variety of driven rods, buried pipe, metal plates, foundation steel, etc... provided that they are all boned together to ensure that no significant voltage differences can develop between parts of the grounding system.) If you build two or more grounding systems, there is very real possibility that there will be undesirable or even dangerous voltage differences between the two systems, particularly when some fault current exists.
For example, if your antenna comes in contact with power or is charged by atmospheric static, current will flow to ground through whatever path you have provided. If that path has just a few ohms of resistance, a significant voltage drop will be present so long as the fault current exists. Driven ground rods are almost always going to have significant amounts of resistance to earth. The last two I drove and checked had about 1500 ohms of resistance to the surrounding earth. The two in parallel were about 800 to 900 ohms. That means that if a 1 amp fault current were to occur during a nearby lightning event, I could expect there to be about 800 volts between the antenna and nearby soil.
As you can see, an antenna system connected only to these two isolated ground rods could be energized with a hazardous voltage relative to an electrical system elsewhere on the property. If I ran the coax to a TV that is connected to power that is protected by a separate ground, there could be a large voltage difference between the coax and TV chassis.
In my case, the two rods were to serve an outbuilding power sub-panel. The supply from the main building includes two 'hot' wires, a neutral and a grounding conductor that originates from the main electrical service panel. So I'm not depending on the ground rods at the out-building to provide the primary connection to earth... They simply add to the already existing grounding system thats present on the premise and which is also tied (bonded) to the utility companies grounding network via their service drop to my electrical service panel. The resistance of the grounding conductor between the out-building sub-panel and main service is less than 1 ohm, so even if current flows, the voltage drop is far less than if I had relied on the soil to connect the grounding system in the out-building and the main building.
So my advice would be that if you choose to add a ground rod near the antenna (or anywhere else on your property for that matter), run a #6 AWG copper 'bonding' wire to the existing electrical service ground.
You also have the option of running a mast ground all the way to the existing electrical service ground... AND, run the coax to the electrical service ground where you would install the coax grounding block. These connections can be done with #10 AWG copper (or heavier if you wish).
(I keep the coax outside the building until it has passed through the coax grounding block. I locate the coax grounding block close to a bonded ground rod so that the wire from the block to the grounding system is as short as practical.)
http://forum.tvfool.com/showthread.php?t=901 (see post #20)