If you have no common wire you have no return path for current so that you can power a consuming device. Original thermostats were entirely electromechanical switches with no power requirements.
The Nest gets around the common requirement by cheating - it charges a rechargeable battery by leeching current into the signal wires. If the endpoint is a coil on a relay or solenoid, this usually causes no problem, it often causes no problem if the relay is solid state, but on some equipment that return current will switch on the equipment when it tries to charge its own battery. In other cases it won’t charge at all. Usually it’s quite clear if this happening as the equipment will be cycling even with no call for heat.
Yes, what happens is that the power draw is large enough to cause the heater to see that as a demand for heat. There are little converters for this that ensure that the thermostat has enough power to run and the heater only sees a potential free switch when there is actual demand. Google 'Nest Thermostat C-wire adapter'. This is especially convenient if you'd otherwise have to run new wiring which can require a lot of work.
The Nest gets around the common requirement by cheating - it charges a rechargeable battery by leeching current into the signal wires. If the endpoint is a coil on a relay or solenoid, this usually causes no problem, it often causes no problem if the relay is solid state, but on some equipment that return current will switch on the equipment when it tries to charge its own battery. In other cases it won’t charge at all. Usually it’s quite clear if this happening as the equipment will be cycling even with no call for heat.