A quick search through the internet will give you the same basic responses: yes/no/depends on the story. Obviously only the last one is right since there have been a plethora of prologued publications. But the simple response—does the story need it/can it be told without it?—seems too arbitrary for me, like some sort of political talking point.
Caveat: this is primarily aimed at initial prologues, i.e., in a series.
Prologues often occur in stories that are fundamentally about something deeper and overarching to the protagonist’s story. For example, The Wheel of Time, by Robert Jordan and Brandon Sanderson is about the eternal struggle of Good vs. Evil, as avatars of each are continuously reborn throughout time to do battle, and this series is about one particular rebirth. The prologue sets this epic backdrop which is what the story is fundamentally about: “This war has not lasted ten years, but since the beginning of time. You and I have fought a thousand battles with the turning of the Wheel, a thousand times a thousand and we will fight until time dies and the Shadow is triumphant!” In the Magister Trilogy, by C.S. Friedman, we are introduced to a type of magical power, which is what the trilogy is fundamentally about. George R.R. Martin’s A Song of Ice and Fire is about the dead vs. the living and the wall that divides them, which is exactly what we see in the prologue. Prologues often tell a story that lays out the foundation for the ostensible story to be told upon.
Some stories need a prologue the way a book needs words to be read. They don’t just provide a necessary backdrop or foundation, they are essential for understanding the story itself. The first books in Jordan’s and Martin’s series could both be read without reading the prologue, but doing so would compromise the quality of the story. Eschewing the prologue in Friedman’s Magister Trilogy is would leave the reader completely lost. In the first two examples, the tale of Good vs. Evil isn’t just the backdrop, it’s the foundation upon which the story is built; Friedman’s trilogy is about a novel twist on vampires and understanding the how magic system works is a fundamental necessity to understand not only the story but the motivation. The first two are like walking ten minutes late into a drama; the latter is like walking ten minutes late into a mystery.
That being said, if you have a story that is deeper than the protagonist’s ostensible story it doesn’t mean you necessarily need a prologue. Take it out and see what happens. As I was writing the first book in my series War of Ages, I took out the prologue and couldn’t make the story work. In a different manuscript, I started with a short story that could definitely be used as a prologue. It sets the entire back story to the ruined world the reader is introduced to, but it’s not necessary for the progression of the story. In fact, including a prologue in this particular novel would ruin the underlying mystery as it unfolds throughout the story.
A quick note on writing prologues: they aren’t backstory dumping grounds. They are to set the foundational conflict for the story, the ‘what’s it about’ beyond the protagonist’s story, not to extrapolate cultural intricacies between societies, etc. A prologue should be stories within themselves, short and to the point. Culture etc. should be explained as the story unfolds.
So, how do you know if your story needs a prologue? If you already have one, take it out and see how the story feels. Give a couple of beta readers your manuscript without the prologue, and then once they read it, give them the prologue and ask their opinion. If your story doesn’t have one and you don’t know if you need one or not, write one anyway and ask your beta readers after they’ve read it without. Whether or not to have one completely relies on the story. Is it about a kid and the bond he forms with his two hounds or is it about living in the Great Depression as told by a kid who bonds with his two dogs?