Not super secret, mind you. Some rolls you'll roll without intending to reveal the result, but if someone accidentally sees the result, it's no big deal. I'd classify monster attack rolls this way.
But most rolls should be secret. Not just because of suspense, although that's very important. But because, as I'm sure you've heard me rant before, players and play should not be focused on mechanics. Mechanics are for GMs. They aren't important.
So players get to roll their own saves, because they may already know their target number anyways and keeping the roll a secret is pointless, since it won't reveal any information. They can roll their own attacks, although I'm in the "roll high on 1d20" camp rather than telling players target numbers. They can roll for many situations, where they know what's happening and know the approximate odds, like "The narrow ledge near the waterfall is damp. If you cross it, roll a d6. On 5+ you slip."
But reaction rolls are secret. Turn undead is secret (especially the way I interpret it, since two Very Bad results on Turn Undead means the cleric has angered their god or had their faith shaken.) Monster saves are secret (they are clues to monster level.) And definitely, any search for something that might or might not be there, like secret doors or listening at a door, has to be secret.