The service is the meter main with the two 200A disconnects. That is where the ground electrodes should connect and not at any down stream panel.
You need to feed each 200A panel with 4 wires and make sure the neutral and ground are not bonded in either down stream panel. This may be a pain in the existing 200A panel if it is old and is a bit short on neutral and ground bus terminals. Could also get ugly if the wires are too short and you need to wire nut on some pigtails.
Wire size is determined by the source feeder breaker, which is 200A in this case (unless you want to use feeder tap rules, and you probably don't). So for 200A, you use 3/0 copper (with #6 equipment ground). If going with aluminum, you use 4/0 if the calculated load is 180A or less, or 250 KCMil if you're pushing the full 200A. Use a #4 equipment ground if using aluminum.