What is a "complete secret"?

Question

“Good afternoon, Harry,” said Dumbledore.
Harry stared at him. Then he remembered: “Sir! The Stone! It was Quirrell! He’s got the Stone! Sir, quick —”
“Calm yourself, dear boy, you are a little behind the times,” said Dumbledore. “Quirrell does not have the Stone.”
“Then who does? Sir, I —”
“Harry, please relax, or Madam Pomfrey will have me thrown out.”
Harry swallowed and looked around him. He realized he must be in the hospital wing. He was lying in a bed with white linen sheets, and next to him was a table piled high with what looked like half the candy shop.
“Tokens from your friends and admirers,” said Dumbledore, beaming. “What happened down in the dungeons between you and Professor Quirrell is a complete secret, so, naturally, the whole school knows. I believe your friends Misters Fred and George Weasley were responsible for trying to send you a toilet seat. No doubt they thought it would amuse you. Madam Pomfrey, however, felt it might not be very hygienic, and confiscated it.”
(Harry Potter and the Sorcerer's Stone)

In the context, ‘a complete secret’ seems to mean ‘an incident already well-known’. But I don’t find any explanation in dictionaries. What does it mean?

Answer 1

The sentence represents a bit of irony on Professor Dumbledore's part. By “[it is] a complete secret” he means that what happened is, officially, entirely a secret. “So, naturally, the whole school knows” suggests that at Hogwarts the deepest secrets are soonest known, because so many are interested both in hearing and telling them.

Answer 2

It's meant to elicit a laugh or chuckle. Evidently, gossip flies fast at Hogwarts, as jwpat7 indicated.

As you know, Hogwarts is an academy filled with school-aged children. Like in so many other institutions with children that age, nothing gets kept secret very long. Someone will tell a friend or two, and those friends will go tell a friend or two, and so on, and so on.

In today's parlance, Dumbledore might have said:

What happened down in the dungeons between you and Professor Quirrell was supposed to be kept secret, but, naturally, it went viral, and now the whole school knows.

In the context of the original, the word complete doesn't really change the meaning of the sentence at all; it merely acts as an intensifier, much like the word very would.

Answer 3

A complete secret is only known by the direct participant(s) of an event. It you stole an exam with a friend and cheated on a test, it would be a complete secret if only the two of you know about it. On the other hand, it would be a "secret" (not a complete one...) if you also told some best friend or person of trust.