Thanks for the helpful replies so far.
I understand that my home network firewall is the real gatekeeper, but I don't know whether the Internet-related features of DirecTV would require me to open up any incoming ports through my firewall to the DVR or receiver. Currently all incoming ports are closed on my firewall.
Perhaps I am being unnecessarily spooked by security concerns around the Internet of Things - home devices such as thermostats, refrigerators, surveillance systems, lights, door locks, etc. When such devices are made visible on the Internet so a homeowner can monitor and control the home from afar, the home becomes vulnerable to hackers and malware. Such intrusions might be benign (Kilroy was here) or could be very destructive (shutting down the furnace so all the water pipes freeze). I recognize there is limited destruction that could be done on a DVR - deleting titles I recorded or recording titles that might hurt my reputation with certain people - but I would prefer to know that path need not be opened at all.