Over many years of playing D&D and reading OSR blogs, I've developed some pretty strong views on DMing. Many of these thoughts have to do with transparency at the game table. DM screens have long been considered an essential piece of equipment by many, so obviously a lot of other people consider the transparency vs. secrecy issue important. The truth (as I see it) is that screens are often more of a hindrance to good gaming than a help, and most of the stuff that goes on behind them ought to be done in the open, for all to see.
DM transparency fosters player trust. The less you hide from the players, the more they can have confidence that you're not secretly screwing them or coddling some particular favorite character, or the entire party. They'll become better players when they can clearly see how their decisions play out through the impartial application of rules and dice rolls.
But transparency also enhances the experience of the DM, too, by removing some of the temptation to fudge dice rolls in pursuit of a predetermined narrative. If you want to tell stories with total narrative control, write some form of traditional fiction. D&D is a game, not a novel, and a large part of the fun for both players and DM is seeing how things turn out in often unforeseeable and unexpected ways. DMing is a different skill set to writing. It's learning to roll with the punches, to adapt on the fly as you incorporate the results of the players' choices, the rules of the game, and the roll of the dice into the emerging chain of events. You can't develop the skills of a good DM by exercising the control of a novelist, and so you deprive yourself and your players of a lot of the uniqueness of playing D&D vs. writing or reading a book.
Below are some of my guidelines for DM transparency -- all of which, at some point during my DMing career, I was in the habit of routinely violating, and thus providing me plenty of first-hand experiences on which to reflect over the years.
1. Roll dice in the open. Whether it's attacks and damage, saving throws, or wandering monster checks, there's no reason why the players shouldn't see the roll. If it's something that a player really shouldn't know immediately, leave the die alone after the roll if possible, and show it to the players after the situation is resolved. (Even then, it's possible to handle rolls such as a thief's hiding and sneaking in such a way that no secrecy is needed -- for instance, roll only when someone is in a position to notice the thief, and thus the result is immediately known to the player.)
2. Don't roll dice unless you intend to abide by the results. Rolling and then disregarding the result gives the players the correct impression that the dice don't really matter; you're going to overrule them whenever they don't produce the result you want. If the outcome of some proposed action is abundantly clear to you, don't dilute the authority of the dice by making a pointless roll. Own that decision, and announce it without touching the dice. "DM fiat" like this is perfectly acceptable in many instances, such as reactions and wandering monster checks, and at least less problematic in cases when you just want to cut the players a break. (Frank Mentzer gives an example in his Basic Set of a character with 3 hp being attacked by a monster using a 1d8 damage weapon, advising the DM to roll the dice and then announce 2 points of damage, whatever the actual result. I think this is terrible advice. If you must be lenient to a new player, make it clear that's what you're doing, and leave the dice out of it. Own it!)
Also, don't roll dice just to make players nervous. Don't roll for encounters if you intend to just choose whether one will occur or not, or indeed, have already decided it won't.
3. Be clear with the players about the odds of success when they want to try something not explicitly addressed in the rules, at least as far as their characters would be able to observe and discern. "You estimate about a 75% chance of leaping over the chasm." "If you try that, you'll need a roll of 5 or better on 1d6." "No, there's no conceivable way that could work, and you'll almost certainly die if you try it." Then stick to that assessment. "You're not sure if that would work or not" is an acceptable answer in cases where the characters would truly be unable to assess the odds, but that should be an uncommon scenario, and a strong signal to the players that they might need to seek more information.
4. Don't make it arbitrarily impossible for players to gain information. Allow them to discover it when their actions would reasonably be expected to reveal it, even if their learning it spoils some big plan of yours or takes things in a direction you didn't anticipate. If they search the right area, they find what's there to be found. If they ask the right question, and the NPC would know and has no motive for withholding the answer, give it to them. If you feel players haven't been specific enough in describing their actions, prompt them to be more specific -- "How exactly are you examining the wall? Visually only, by prodding with a stick, or touching it with your hands?" -- but don't pixel bitch and don't expect them to read your mind.
5. Don't play "Gotcha!" Provide clues and foreshadowing of danger, so players can make informed choices. Clues don't always have to be blatant and in-your-face, but it should be possible to notice them with reasonable efforts and possible to deduce their meaning and significance. Number 4 is very much in effect: If players do something that would reveal the presence of a trap or hazard, it's your duty as DM to allow them to find it, not create rationalizations for not revealing it because you're so proud of your design and would feel slighted if you don't get to spring it.
6. Don't cut corners, and don't "guesstimate" after the fact. Do the math! I've known DMs who will look at dice rolls and decide ex post facto whether or not they succeed, which is a gross misuse of dice. This could be due to simple laziness or to an unspoken desire for a certain amount of "fudge factor" to allow for DM fiat at the margins. Either way, it's bad form. Could be considered a corollary of Number 2 above: Don't have players roll dice if you're not going to abide by the results, objectively and consistently. As a sub-corollary of this point, don't fudge monster stats in the middle of a fight to nudge things in the way you want them to go. You can faithfully count every pip of the dice rolls, but tweaking ACs or hit points en media res is having exactly the same effect as being fuzzy with the pips. It's just moving the goalposts in a different way, and it's a dick move.
In summation, there are some things that are rightly and properly hidden from the players, and it is the DM's prerogative alone to know in advance, for the sake of an exciting and entertaining game. By all means, do keep setting details (e.g. dungeon maps, the locations of monsters and treasure, the monsters' motives, the villain's plans, shocking details about the setting's history, the identity of the true heir to the throne, and so forth) secret -- but only until the players discover them in play. All of it ought to be discoverable in play, and everything that happens to the characters should be transparent. Players ought to see the rolls that determine their characters' fates, including all the attack rolls, damage rolls, and saving throws, both their own and those of their opponents.
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