If you want a less dark explanation, I think the toys are powered by both individual and collective imagination. Really popular toys like Buzz gain sentience immediately, because there are masses of children already thinking of them that way so much so that proximity isn’t needed. Non-toy items like Forky can gain toyhood and sentience from a concentrated dose of imagination. Most toys would be somewhere between those two extremes.

You could also go a step further and say collective imagination results in a lower quality sentience, explaining all the Buzzes, green aliens, etc. starting out less self-aware.