If players sneak around enemy NPCs the NPCs make a listen check. But what about when an enemy NPC is sneaking up on players? Do you tell all the players to make a Listen check? And if yes, how do you deal with the fact that this reveals to the players that something is sneaking up on their characters?
You have two good options here:
Roll for them
Don't have them take 10, as this gives them a statistically worse chance of succeeding than rolling. Also be sure you know exactly how much each character adds to Spot, Listen, Sense Motive, Search and whatever other checks you are handling this way. It is likely that characters who care about these rolls add a fair bit more than just ranks+attribute bonus, so be aware of that. When you start to do this and at the beginning of all future games in which you plan to do this (which will likely be all of them), let your players know that you will be making some of their rolls for them secretly. Explain what kinds of rolls you will be doing this with and why. Explain that while you make an effort to understand the mechanical details of their characters, they know their characters best and, since they will not be able to remind you of any unusual special bonuses they have when the rolls are being made, they should make an effort to do so beforehand. Ask them to keep you appraised of the relevant totals for each kind of roll and any special situational modifiers that might apply only to their character (+3 to spot in bright light, for example). When the rolls come up, roll them secretly. In general, you should not let the players know what you are rolling for or even what kind of roll it is. You should also make sure you roll dice for the weather or some other task that means that you will be rolling dice frequently for reasons that are both not immediately obvious to the players and not some kind of perception check.
Ask for rolls with no consequences
Obviously if a player rolls poorly on a Spot check he will suspect something is sneaking up on him. However, while rolling poorly will still make players nervous, if you regularly ask for Spot checks when there is nothing to find, especially when the players believe themselves in danger, your players will not be able to know something is up when they roll badly, and will get over their fear of secret monsters eventually. Until they next fail a Spot check anyways. Again, you should let your group know that you will be doing this before implementing this, and you should not call for such checks so frequently as to bog the game down. You should also be aware that natural 20s on these fake checks may make some players slightly resentful.
Would players be tempted to do something out of character that might spoil the game for fomeone? If not, let them roll. Are you (almost) sure they will resist the temptation? Let them roll. Rolling is fun, so unless a roll spoils some bigger fun, let the players roll.
Most of my passive perception rolls (different system, but shouldn't matter) are done by the players, because failed rolls don't spoil the game. The key is that the PCs will learn the important think anyway and sucessful roll means an advantage or failed roll means a complication. This is not always possible; if not, let them row X times in the beginning of the session and just check the pre-rolled numbers, perhaps in order set by some algorithm unknown to the players.
Typical situation where the players should roll is encountering a trap in the middle of a tunnel. Should the PC searching for traps fails the roll, it's time for saving rolls, not for arguing that the PC wants to stop just before the trap and announce: "I have missed a perception check! Let's prepare for a trap or an ambush!" On the other hand, if the trap is on a door the PCs might open or not, the situation is trickier: unless the players roll for every door, or are very good at separating IC/OOC knowledge, use a pregenerated roll.
Another example: ambush! If the enemies are going to attack the PCs anyway, then succeeding or not will only determine initiative - there's no more knowledge the player will get from a failed roll. But if the enemies just follow the PCs and remain hidden, I would use the pregenerated roll.
One more example: PCs are investigating a murder. While examining the place of the murder, they should get the clues anyway, but should they miss the roll, it will take long enough for the murderer to leave the city (and take advantage to the final riding chase scene). Or if they don't work for city guards, the guards will arrive just after finding the clues and mistakenly accuse the PCs of trying to destroy the clues leading to one of them. It is quite different style, but I like it more than explaining players that repeated examinations after initial fail won't show anything new.
TL;DR: if failed roll means a complication, not missing the information completely, always let the players roll. Otherwise let them roll in the beginning of the session and then just use those rolls' results.
No. I think the best way to handle this type of situation is to borrow from later editions and/or flavors of D&D. Make it a passive check!
Simply let them all take 10 on their d20 rolls and add their bonus to it. If they succeed, then start giving them clues. You can handle Spot the same way. Cunning players will pick up on this and start making explicit calls for Listen/Spot checks on their own in appropriate places like dungeons, caves, etc. if they think they have a better chance of succeeding with a roll of the die.
You might think that the better option is to roll for your players because this gives them a better chance statistically to beat a high DC. I disagree. It makes sense for a passive check to be less effective against high DCs because no person can be hyper-vigilant every second of the day. Also, you need to be able to reward players who call for explicit checks at the appropriate times. They deserve to have a better chance to spot a sneaking Shadow Thief if they are looking for him.
This is a typical example for the separation and (at the same time!) inseparability of in character (IC) and out of character (OOC) knowledge.
Ideally, your player's will strictly separate these two. In practice, this is often not the case, regardless of whether it's intentional or not. If you know somethings up, it's really hard to not let it influence the decisions your character would make. Failure to do so might be called "bad roleplaying", but it most certainly does not make you a bad roleplayer. I know few people who can do it well.
Let's look at the options:
1) Letting them roll
OOC, everyone knows something is up.
IC, characters that fail their check won't know a thing, those that succeed have noticed something, and their players should be notified OOC.
Going one step further, you can notify the players of what they did or did not notice openly, or separately. Both options give the "succeeders" all the information they are entitled to, but the former gives the "failers" now get even more OOC knowledge on top of just "something is up". The latter however might be a bit tedious, especially if relaying the information to the other characters does not cause any problems by itself (e.g. alerting the enemy as well).
2) GM rolls for everyone
One alternative here is rolling the dice for your players. Have them tell you their bonus for "passive" skills (like listen, spot, sense motive, etc.) beforehand and roll for every player, and noone is the wiser.
This doesn't totally solve the problem though. First of all, rolling #Characters dice will be noted, just as the move silently rolls will get noted. Since the GM doing anything typically makes life harder for the party, your players will be on edge regardless, only with a little less information (What did he just roll?).
You can alleviate this problem by pre-rolling a lot of d20s and noting the results on a piece of paper. This can be done either before the session altogether, or periodically during the session, or when the game is on smoke/toilet break. This way, there's no clanking to be heard or the clanking of dice is decoupled from immediate danger. Or at least that's what your players think, dun dun duuuun...
However, if only a single player doesn't fail, you have to relay information. Even if you pass a note, everyone knows somethings up. So in most cases, stealth rolling only delays the inevitable, though it can be useful in cases where you don't expect any of the characters to succeed, barring very good luck.
3) Passive opposition
This is largely similar to the GM rolling, but without the rolling part. Instead, just add 10 to your players' bonus and call it a day.
This also means that luck is completely out of the equation, on the players' side. If anyone hears it, the character with the highest bonus will hear it. Whether or not this is a problem, depends on your group. You can prevent this by rolling the move silently against every player, but that puts you right back with rolling a lot of dice...
All three methods have their merits and use-cases. None of them are particularly stealthy on the GM's part. Even if you pre-rolled or use passive opposition, your players will notice you staring at your notes.
There is one last thing I should mention in the line of this answer:
Rolling dice is fun
This depends largely on the group/your players. Personally, for me, it an integral part of the fun. Something happens, roll the dice. This alone puts option 1 to the top for me. Getting to roll the dice negates most of the downsides for most cases (cf. last paragraph of 2)
[Anecdote] I have played a debuffing Witch in Pathfinder, and even though I like the character, he is not fun to play, since I never get to roll any dice in combat. [/Anecdote]