Like, in Episode VI, from imsdb:
LUKE: I've accepted the truth that you were once Anakin Skywalker, my father.
VADER (turning to face him): That name no longer has any meaning for me.
And if I remember correctly, in the movies Darth Maul hasn't been called by his real name.
While, in the movies, Darth Tyranus usually goes by his name "Count Dooku" and Sidious by his name "Palpatine".
At what considerations do a Sith decide whether they go by real name or Sith name? (in-universe)
The main difference between Sheev Palpatine and Count Dooku as a set, and Anakin and Maul as another set, is circumstances:
The former pair were politicians with their "muggle" personas being their public faces. They had pre-built political careers, and used them as disguises and masks; and political tools.
Maul isn't a good example as he had no name and no past to speak of different than his Sith idenity. He was trained as a Sith apprentice since early age by Sidious - he was given to him as an infant by his mother (src: "Darth Plagueis" as well as "Clone Wars").
Vader was the reverse of Dooku and Palpatine:
nobody was supposed to know his past or real person. The fact that he used to be a high-profile Jedi was to his detriment.
He wanted to forget his past, as it was painful for him:
Ryloth filled the viewport as the ship descended. Seeing the mottled browns of its surface, the smears of green and tan, dredged memories of other times up from the sludge of his distant past, names he rarely thought of anymore. Anakin. Mace. Plo Koon...
The shuttle hit the atmosphere too sharply and skipped and bounced ... Flames from the friction of atmospheric entry sheathed the ship. Fire surrounded them. Fire.
Mustafar.
Obi-Wan.
He used his ever-present anger to burn away the memories, but the charred husks of the past clung to the forefront of his consciousness.
Padmé.
(Disney canon novel "Lords of the Sith" by Paul Kemp)
....
“And I see you, Master. You think I long for the past when I see it in visions, but you’re wrong. I don’t long for it. I think of it and the man I was then and regard it all with contempt. And the only thing that makes it tolerable to ponder is that it ends with me here, in this armor, with you. I feel no longing. I feel no regret. My memories feed my anger and my anger feeds my strength and so am I able to serve you, and the Force, better. Your doubt…”
To top that off, Palpatine seemed to be very intent on suppressing Anakin's former self. Letting go of your old personality is part of that.
A Sith decides to use his real name or Sith name depending on which would be more effective and useful for the Sith Order. Different individuals have different skills, so some Sith Lords are more effective as overt Sith whereas others are more effective as hidden Sith.
Palpatine/Sidious was a skilled politician who instigated various crises in order to gain power.
The first such crisis was the Trade Federation's invasion of Naboo. As Darth Sidious, Palpatine ordered the Trade Federation attack Naboo. Meanwhile, Palpatine was Naboo's Senator in the Republic Senate. Palpatine used the Republic's failure to protect Naboo from the Trade Federation to unseat the Supreme Chancellor at the time (Finis Valorum) and acquire the office himself. The fact that Palpatine's home planet was invaded no doubt helped him gain sympathy votes for his election as Supreme Chancellor.
Once he became Supreme Chancellor, he instigated the Clone Wars. The ongoing emergencies of the Clone Wars allowed him to acquire new powers for himself as the Republic's Supreme Chancellor. Finally, at the close of the Clone Wars, the Jedi discovered he was secretly a Sith Lord and attempted to arrest him. Palpatine's face was disfigured when the Jedi attempted to arrest him, but he was able to label the Jedi as traitors and order them executed. Palpatine was able to proclaim himself Emperor while portraying himself as a kindly old man brutally attacked and disfigured by the power-hungry and traitorous Jedi.
Palpatine was far more effective as a politician with a hidden Sith identity than a Sith combatant. He was able to gain and keep power primarily through political crises which garnered sympathy for him. He would not have gained such sympathy nor any political office in the Republic had he publicly used his Sith name and overtly presented himself as a Sith Lord.
Dooku's primary purpose as a Sith Lord was to serve as the leader of the Confederacy of Independent Systems (CIS) in order to instigate the Clone Wars. As previously mentioned, Palpatine used the Clone Wars to gain more power for himself. Dooku was an effective leader for the CIS since he was charismatic, wealthy (he was the Count of Serenno), and politically well-connected (he was a former Jedi Master). As a political leader, it was important for Count Dooku to emphasize his noble heritage from Serenno rather than openly call himself the Sith Lord Darth Tyranus.
On the other hand, Maul and Vader were both more effective as terrifying, overt Sith Lords.
Maul's tattoos make him look evil, making it impossible for him to pull off the "kindly old man" persona used by Palpatine or the dignified, noble persona used by Dooku. Maul therefore embraced his persona as a fearsome Sith warrior. Maul concentrated on learning martial arts and mastered the double-bladed lightsaber, making him particularly well suited for assassinations and lightsaber duels.
Vader was similarly terrifying due to his mask and armor, his use of Sith powers like Force Choke, and his similarities with General Grievous:
It was that genuflecting obedience, the steadfast devotion to execute whatever task the Emperor assigned, that had given rise to so many rumors about Vader: that he was a counterpart to the Confederacy’s General Grievous the Emperor had been holding in reserve; that he was an augmented human or near-human who had been trained or had trained himself in the ancient dark arts of the Sith; that he was nothing more than a monster fashioned in some clandestine laboratory. Many believed that the Emperor’s willingness to grant so much authority to such a being heralded the shape of things to come, for it was beyond dispute that Vader was the Empire’s first terror weapon.
- Tarkin, p. 73
In Vader's case in particular, his (former) Jedi identity as Anakin Skywalker conflicted with his Sith persona. Anakin was a hero for the Republic in the Clone Wars, but Vader was the Empire's enforcer and terror weapon. Vader therefore had to hide his identity as Anakin.
Furthermore, both Vader and Sidious wanted Vader to forget his former identity as Anakin Skywalker. Vader wanted to forget his painful loss of Padme and therefore needed to forget his former life as Anakin. And Sidious knew that Vader needed to forget his life as Anakin so that Vader would not consider Luke to be his son and would remain loyal to the dark side. Indeed, Luke's response to Vader's claim that "Anakin" no longer had any meaning for him is:
It is the name of your true self. You've only forgotten. I know there is good in you. The Emperor hasn't driven it from you fully. That is why you couldn't destroy me.
Dooku and Palpatine were both trying to hide the fact that they were Sith. They had secret identities, so they were using two names.
Anakin, at the time of his conversion, had no one left to hide from. In addition, due to his injuries, there was no way for him to be seen and pretend to be a Jedi any longer.
I'm going to answer this question in four parts, one for each person you listed, but first you need to understand something about the Sith. When Sith are inducted into the order as apprentices and are given their new names they abandon their old life and will often talk about who they were before they were Sith by using their old name but those names don't apply to the Sith. At least not in their minds. But the Sith of the rule of two would almost always use their old names in public. But, as I said, each person you listed has a different reason for the name they go by.
Palpatine (Darth Sidious) - He used his old name for the sake of deception and he only used it in public but whenever he was talking to a minion of his he would use his Sith name.
Count Dooku (Darth Tyranus) - Used his real name because he was already a public figure so using his old name was just much more convenient although he didn't really need to since the Jedi already knew he was a Sith, after he'd faced them in Ep. II.
Darth Maul - I don't know of any Cannon information that would explain this but in the Legends novel, Darth Pleugies, Sidious found Maul as an infant and raised him as a Sith so Maul didn't have any other name.
Darth Vader (Anakin Skywalker) - By the time Vader became a Sith the need for secrecy was gone and the need to frighten people had arisen like I said earlier Sith do not consider themselves to be the same person they were before so to Vader, Anakin was a totally different person and it was more beneficial to the Empire for him to use the terrifying and mysterious name of Darth Vader rather then the less threatening and more well known name of Anakin Skywalker.
I think it depends on their existing reputation and what they want their image to be. It made sense for Palpatine, as a well-known career politician to keep his name as Emperor, as a symbol of a new, but still recognizable order to former republic citizens. Furthermore, it would have made sense for Palpatine not to widely use his Sith name in order to maintain his illusion of being a servant of the people, and to give an impression of distance between himself and the Sith he employed.
Conversely, Vader had plenty of reason to switch to his new name as part of a larger transition from his former self into the monster he would become. It was likely as much an internal change (turning his back on his past affiliations and way of life) as an external (distinguishing himself in the eyes of the universe from the Jedi he once was).
Count Dooku had no need for such distinction, as he had already returned to his home and life as Count of Serenno, after abandoning the Jedi, and his training as a Sith was more an extension of his own native lust for power than a transition from one way of life to another. Also, like Palpatine, Dooku was already well known and there was a power to his own name and title that was not hindered by his former association with the Jedi.
As for Maul, he was taken and trained by Palpatine at a young age, so it's likely that Darth Maul was the only name he really identified with by adulthood.
So, using these cases as a jumping off point for considering the in-universe motives for some Sith to use their Sith names and some to keep their prior names (in addition to their new ones), it seems the biggest factors are whether or not the new name can give them a net gain of power and influence, in addition to whatever personal factors may be at work related to baggage from their pre-Sith lives.
