The thermal paste from the factory is non-conductive at the time of installation. It COULD become conductive over time but that would take a loooooong time. The reason manufacturers use non-conductive paste is for part safety. I've seen some seriously ugly blobs of stock thermal paste in laptops
To add to staggeredsix : It's not conductive, but even if it were, there's nowhere near enough on stock coolers to spill over the CPU as they're usually screen-printed. Personally, I replace it with Thermal Grizzly or Gelid right off the bat, I prefer good, reliable thermal paste sources to non-descript stock paste. In laptops it's a different story and it's more often than not, absolutely horrible quality. Re-pasted my laptop after 1.5 years, TIM was cooked to the point it just flaked off!
Also best to replace it every 6-12 months if you have consistently high workloads, 2 years tops for normal use with a good paste. That applies to GPU as well.