There's been a couple schools of thought. There is a German archaeologist named Vinesz Brinkmann. He found traces of paint have indeed on quite a few pieces of statuary - like this bust of Caligula. [Here he is in a German museum, repainted according to the flecks that remained.] His hypothesis is that "plain white wasn't considered ideal until the Renaissance." Here's that article.(https://en.wikipedia.org/wiki/Roman_sculpture#/media/File:Caligula_Rekonstruktion_Polychromie.JPG)

There's also a very brief little Q&A article from Mary Beard, the famous Roman historian, on the topic. She, on the other hand, wrote that she's more skeptical and thinks the majority of statues were likely unpainted, because there's a lot of ancient writers who referred to the white colour of statuary. Here's Dr. Beard's brief little article.

So honestly, we can't really say when/why colour would have been selected or not, but we're pretty sure that both painted and unpainted existed because of those texts Beard mentions. Since so much time has passed, not every statue still has paint flecks, and we can't tell if it's just lots of wear or if it was never painted in the first place. In terms of differences across time, it was only around 100 AD that the pupil and iris started to be carved into the statues, which suggests it was more common to paint busts or faces before then. But they could easily have still painted it afterwards.