Landscape lighting system only puts out one volt
The usual cause of this kind of symptom would be a short, likely an internal short in the power supply, but if you're seeing the power supplies both do the same thing and both work on one side of the house, it must be some kind of "feature."  

I have not used a power supply that works like this;  most low voltage DC power supplies will show full voltage on the load terminals with nothing connected.  As mentioned in the comments, maybe it's a safety feature.  (If so it's a little surprising to me, since 12V is pretty safe, I could see this making more sense with 120V...)  

But going with this presumption that this is a safety feature, where the transformer cuts voltage in a no-load condition, my first guess would be that implies one (or maybe even both) of the conductors between the power supply and the first light are broken.  Maybe they got damaged somehow while decorating, maybe someone drove a stake in the ground and hit a conductor.  

If the wires were in tact to the first light, you would have a load.  If the fault was downstream from the first light, you'd get at least one or some lights working.  

The simplest way to test this would be to run a test cable across the yard from the power supply to the first light and see if everything works.