The individual would pull their dark powers into themselves and adopt a mask of insignificance. This projected illusion would enable a Sith to hide in plain sight whenever they would go out in public.
Palpatine was in all actuality a very deformed Sith Lord from decades of using the Dark Side. To hide this, he used force mask, which also hid his yellow eyes.
As a result, Palpatine's face changes in Star Wars: Episode III - Revenge of the Sith because his dark side-induced mask was broken by the Force Lightning rather than the Sith lord actually being disfigured by his own attack.
Younger or less experienced Sith cannot hide their eyes from others because their dark side emotions are too new to control. For instance, a newly christened Darth Vader went back and forth between yellow and his normal eye color.
However in order to hide his corrupted looks from the Jedi - and the entire galaxy - Sidious used a technique to mask his looks and to appear normal. When Mace deflected Palpatine's own Sith Lightning back to Palpatine the mask collapsed revealing Palpatine's true face.
1 Darth Sidious
There has been no Sith, as powerful, meticulous, nor as evil as the Emperor himself, Darth Sidious. As previously mentioned, Sidious trained in the Dark Side by Darth Plagueis before Sidious killed Plagueis in his sleep, making Sidious a Sith Master.
Palpatine chose to keep his face as a symbol of the Jedi's treachery and as a way to appear weak to his enemies. The Sith Lord's rise to power based itself on appearing weaker than he actually was, and he continued that until his death.
Though one thing is for certain: Kylo Ren/Ben Solo does not have yellow eyes because he's not "corrupted" by the Dark Side. This isn't a bad answer, though it does go off on a tangent.
As for why his eyes weren't yellow during the march on the Jedi Temple, that could be attributed to inner conflict about his actions. Though Anakin was Palpatine's apprentice at that time, he hadn't full turned himself over to the dark side yet and was just starting to follow the path.
Jedi and Sith alike fear the loss of power, but the Sith choose to harness that fear as a source of strength.
A Sith Lord with the ability to prevent death and create life, Plagueis is the mentor of Sheev Palpatine/Darth Sidious, who, in accordance to the Sith's Rule of Two, eventually betrays Plagueis by murdering him in his sleep, taking his place as Sith Master.
Amedda was one of the few in the galaxy who knew Palpatine was a Sith Lord, and assisted the Chancellor as Speaker of the Senate, helping him manipulate Senators into granting him more and more power.
The Guards were also among the few beings in the galaxy whom the Emperor trusted with the knowledge of his secret identity as a Sith Lord, knowing they would speak of it to no one, not even among themselves.
Moreover, you assume that Master Yoda had all of the answers! It is clear that he did not. He knew only that the Clone Wars would end with a Sith victory, and with the fall of the Jedi. Warning them would make no difference at this point, without knowing where the true threat lies.
Star Wars Confirms Yoda Turned to the Dark Side - and Almost Became the Most Powerful Sith Lord. With the aid of Count Dooku, Yoda could have potentially destroyed the universe if he had gone to the dark side in Star Wars.
In the novel, Dooku remains silent when Sidious tells Anakin to kill him, because he finally realised, that he had been replaced. His master has found a new apprentice, younger, filled with more rawness and power to destroy the Jedi.
Anakin's Third Lightsaber (Red)
Anakin traveled to the river moon Al'doleem to do just that, slaying Jedi Master Kirak Infil'a and confiscating his lightsaber. This was a curved-hilt saber with a green blade. From there, Anakin went to Mustafar to bleed the kyber crystal and turn it Sith red.
Built to replace his father's lightsaber, Luke Skywalker's own lightsaber contained a kyber crystal and emitted a green plasma blade. Its hilt was similar to that of Obi-Wan Kenobi's third lightsaber, as Skywalker modeled his weapon on his late Jedi Master's lightsaber, but simplified some of the design elements.
Rey reforged her lightsaber during the First Order-Resistance War, so when she did it, maybe she picked a yellow lightsaber because she wants to carry on the legacy of the Jedi Sentinels from ages past? After all, the Jedi Sentinels were chosen from the best of the best of Jedi Knights.
The breaks in Ren's mask were meant to be a visual representation of his fractured character. Instead of hiding him as it did before, the mask tells his history, and his behavior is revealed.
A crystal embodies the Jedi as the heart of the lightsaber, and pure kyber has no color until it's bonded to the Force user. This process likely resulted in Rey Skywalker's lightsaber being yellow.
DARK SIDE FRIDAY - Did you know Palpatine feared Qui-Gon Jinn? Palpatine was immensely powerful and he could see the future which made him very confident about pretty much anyone who stood against him. So why Qui-Gon Jinn?
throughout many of the Legends storylines, Vader sought to hunt down every last Jedi. while Palpatine was no longer concerned with the Jedi. Order, in his eyes, Yoda and the Jedi no longer posed a threat to his role. during the early years of the Empire, Palpatine only sent Vader to hunt survivors.
As powerful as Darth Sidious may have been, it's easy to forget he actually seemed afraid of Master Yoda. As seen in Revenge of the Sith, Palpatine initially thought he could deal with the aged Jedi Master - and then, when Yoda proved able to counter the Sith Lord's Force lightning, he fled.