How does the DIRECTV Whole-Home DVR service work within my home? Do I have to have a network cable run to each receiver?

Directv Whole Home service basically allows your to watch recorded programs from one DVR in any connected room provided that it is an MPEG4 receiver (newer HD receiver).

Direct TV Whole Home systems use a SWM (Single Wire Multi-Switch)Hi Def dish and proper switches to allow access to the shows recorded on one DVR from a non-DVR receiver (must be hi-definition).

Wiring is standard RG6 cable that is used in all satellite install and pre-wired houses, however the switch that is used must also be powered separately.

Sorry about the late answer, hope this makes it clear.

So, are you saying that the Whole Home DVR uses the same infrastructure (switch and RJ6 cable) for sharing shows as is used to receive the satellite signals? Meaning, I do not need to run more wires?

Correct, as long as your house is pre-wired with RG6 cable run to each room that you want to put a Directv receiver.

Direct TV Whole Home systems use a SWM (Single Wire Multi-Switch)Hi Def dish and proper switches to allow access to the shows recorded on one DVR from a non-DVR receiver (must be hi-definition). - This means that Direct TV DVRs only use one line when hooked up to a SWM system.