In the multiple campaigns I have been privy to (across DnD 5th, 3.5, and Pathfinder) there have been three different scenarios I've seen.
The upper extreme: the DM/GM rolls everything, the lower extreme: the DM/GM only rolls for NPCs, and the middle: the DM/GM rolls some.
I'm mostly thinking with rolls like Sense Motive/Insight, Perception, Stealth, and other checks.
What are the advantages of each of these methods and what checks should the DM/GM roll in the case of the middle?
The advantages of hiding rolls is that the master can create suspense and drama. When the players are exposed to the mechanics, they can start making decisions that affect their characters in ways that their character wouldn't act if the player didn't have that information available. The advantage of letting players roll is that it makes it easier for a master to focus on the story since players can help take care of some of the mechanics; the game generally moves faster when the players are helping. Also, players generally like to feel in control, so letting them literally decide the outcome by way of rolling is a pretty natural extension of granting that feeling of control (commonly called "player agency").
For games that hide some rolls, the purpose is to avoid leaking too much information. For example, if a player rolls a 1 on Perception, they might start utilizing other skills to confirm if it is really safe, even though the general definition of a 1 would be that the character is blissfully unaware and unlikely to perform any further checks for their safety. Some people even consider this to be cheating, because they're using information they shouldn't have. Responsible role-players might be able to act out their character by separating character knowledge from player knowledge, but casual players probably won't, and take advantage of the information intentionally or subconsciously.
So, while skills are different from game to game, rolls should generally be hidden when it would (a) expose information to the players that would change the character's behavior, despite the character not having that knowledge, and (b) it would heighten the sense of drama when something unexpected happens. You can always choose to expose a roll afterwards, but you can't "hide" it afterwards. If a player is incredulous about a roll, you can still choose to present the evidence if you want to (unless it's a fudge, which masters should reserve only for dire consequences without a deus ex machina save, preferably no more than once or so a game).
My answer is focusing on advantages from a DM's perspective since that is what you asked for. I think you can see the disadvantages as well though from the players perspective. In short. The DM/GM is here to help the players have a good (memorable) time.
This gives the DM/GM fine grained control over what actually happens. In most cases you are controlling outcomes and in short only things you want to happen end up happening. From my experience this usually is a major detriment to players enjoyment of the game however there are circumstances when this could be a positive. If you are playing with players who are more intrigued by the story and having it unfold before them this style of play may not be an issue. A good campaign with great story/ imagery where a DM/GM makes players feel like the dice are controlling this amazing outcome when in reality it's been set is fine in this case.
This really doesn't need much explanation because it is the same as the above example with the exception that players feel more involved and in control of their characters destines. As an DM/GM you will need to put effort into planning for possible side tracks and deviations from what you have planned/ hopped to happen. This is also a good middle ground where newer players can focus more on enjoying the game/ growing their skills and the DM/GM can still control the story fairly closely. Another plus is players are more accepting of outcomes or turns of events because they are the ones who rolled and got that outcome.
IMHO this or the previous playing style is the best way to play. Using this style of play as a DM/GM you need to do a lot of pre-planning and thought about your campaign. Players that are comfortable with playing and enjoy having unrestricted freedom will enjoy this style of play. The advantage as a DM/Gm is that for as much as you plan you can also plan less. This style of playing can take on the feel of massive open world and players are more open to outside circumstances and events intervening with their game. A good DM/GM can use this to control the story line like the first style (DM/GM roles everything) but the players feel 100% in control. For an example check out This Answer to an Unrelated Question and you will see how many options open up for the DM/GM to take advantage of if they just trust the players and work with their backstories/ roles/ etc.
