In searching for a solution to my problem, I came across a few other issues:

In a two-wire installation, if the heater never turns on (eg. during the summer), the battery in the Nest Thermostat can go flat. So the Nest Thermostat will TURN ON THE HEAT, just to charge the battery.

This problem is also solved with a three-wire installation (adding a C wire).

Another problem might occur if you try to put too many Nest Thermostats on the same power transformer. The Nest Themostat uses a lot of power when it is charging its battery, so this adds up. I didn't have this problem, so I can't speak to it.

I only tried this with a 3rd generation Nest Thermostat. I expect other versions are similar, but I haven't tried them.