What year? 6.0 or 7.3?
I'm fairly certain on mine (2006) if I unhook the passenger side I will lose all power, but the driver side I won't. I believe the driver side is only hooked to the passenger battery and not to any of the fuse blocks. So not sure depending on what year you have if its the same or reverse. So that means if the passenger battery is bad, then only the driver side is powering it and when you unhook it, everything will lose power.
Basically test them individually, never does any good to test the batteries together. If you drive short trips, maybe invest in a trickle charger to keep a charge on them.