Well-cast Imperius Curses aren’t visible.
Once the Imperius Curse is cast sufficiently well, it’s very difficult to tell if someone is under it. The only way the Ministry suggests to figure out if someone is under the Imperius Curse is by noticing them acting strangely.
“5. Should you feel that a family member, colleague, friend or neighbour is acting in a strange manner, contact the Magical Law Enforcement Squad at once. They may have been put under the Imperius Curse (see page 4).”
- Harry Potter and the Half-Blood Prince, Chapter 3 (Will and Won’t)
After the Dark Lord’s disappearance, the Ministry had quite a bit of difficulty determining who was forced to serve the Dark Lord under the Imperius Curse and who was intentionally serving him and claimed to be Imperiused as an excuse.
“Years back, there were a lot of witches and wizards being controlled by the Imperius Curse,’ said Moody, and Harry knew he was talking about the days in which Voldemort had been all-powerful. ‘Some job for the Ministry, trying to sort out who was being forced to act, and who was acting of their own free will.”
- Harry Potter and the Goblet of Fire, Chapter 14 (The Unforgivable Curses)
Even after all the Ministry’s effort, they still weren’t always right in their final conclusion on whether someone was under the Imperius Curse, since some Death Eaters were able to escape sentences in Azkaban by claiming they were Imperiused.
“Avery – from what I’ve heard he wormed his way out of trouble by saying he’d been acting under the Imperius Curse – he’s still at large.”
- Harry Potter and the Goblet of Fire, Chapter 27 (Padfoot Returns)
Several of the Death Eaters were able to avoid Azkaban by lying that they were Imperiused - many of the Death Eaters who took part in the Muggle torture at the Quidditch World Cup had avoided by saying they were forced to. That means a large portion of the Death Eaters that day were free because they claimed to be Imperiused and were believed.
“I suppose they had a few drinks tonight and couldn’t resist reminding us all that lots of them are still at large. A nice little reunion for them,’ he finished disgustedly.
‘But if they were the Death Eaters, why did they Disapparate when they saw the Dark Mark?’ said Ron. ‘They’d have been pleased to see it, wouldn’t they?’
‘Use your brains, Ron,’ said Bill. ‘If they really were Death Eaters, they worked really hard to keep out of Azkaban when You-Know-Who lost power, and told all sorts of lies about him forcing them to kill and torture people. I bet they’d be even more frightened than the rest of us to see him come back. They denied they’d ever been involved with him when he lost his powers, and went back to their daily lives … I don’t reckon he’d be over-pleased with them, do you?”
- Harry Potter and the Goblet of Fire, Chapter 9 (The Dark Mark)
Even Dumbledore, one of the most powerful wizards in the series, was unable to tell that Madam Rosmerta was under the Imperius Curse for months as part of Draco’s ineffectual plans to kill him, despite having seen her.
“Very gratifying,’ said Dumbledore mildly. ‘We all like appreciation for our own hard work, of course … but you must have had an accomplice, all the same … someone in Hogsmeade, someone who was able to slip Katie the – the – aaaah …’
Dumbledore closed his eyes again and nodded, as though he was about to fall asleep.
‘… of course … Rosmerta. How long has she been under the Imperius Curse?’
‘Got there at last, have you?’ Malfoy taunted.”
- Harry Potter and the Half-Blood Prince, Chapter 27 (The Lightning-Struck Tower)
That same day, when Dumbledore first left with Harry, he saw Rosmerta, who was Imperiused at the time since she then told Draco he’d be gone, which is why he returned to Death Eaters in Hogwarts. Dumbledore didn’t notice anything suspicious about her, and since this was before he drank the potion in the Horcrux cave, he was in full possession of his senses. Despite that, he only realized she was Imperiused after, when talking to Draco and trying to figure out who his accomplice in Hogsmeade was.
“Lights twinkled from windows over shops and as they neared the Three Broomsticks they heard raucous shouting.
‘– and stay out!’ shouted Madam Rosmerta, forcibly ejecting a grubby-looking wizard.
‘Oh, hello, Albus … you’re out late …’
‘Good evening, Rosmerta, good evening … forgive me, I’m off to the Hog’s Head … no offence, but I feel like a quieter atmosphere tonight …”
- Harry Potter and the Half-Blood Prince, Chapter 25 (The Seer Overheard)
Therefore, in cases of a well-cast Imperius Curse, not even Dumbledore is certain to be able to tell that someone is Imperiused when encountering them.
Badly-done Imperius Curses are visible.
Though it’s difficult to tell in cases of well-cast Imperius Curses, when an Imperius Curse isn’t cast sufficiently well, it is possible to tell by just looking at them. When Harry used the Imperius Curse on Travers and Bogrod, they both had blank looks.
“They’re Imperiused,’ he added, in response to Hermione and Ron’s confused queries about Travers and Bogrod, who were both now standing there looking blank. ‘I don’t think I did it strongly enough, I don’t know …”
- Harry Potter and the Deathly Hallows, Chapter 26 (Gringotts)
When Katie Bell was Imperiused, her friend mentioned that she looked funny. This is likely because she would have been Imperiused by Madam Rosmerta, who was also Imperiused and wouldn’t normally be capable of casting Unforgivable Curses.
“He looked up at Leanne, who had started to shake uncontrollably. ‘How did Katie get hold of this?’
‘Well, that’s why we were arguing. She came back from the bathroom in the Three Broomsticks holding it, said it was a surprise for somebody at Hogwarts and she had to deliver it. She looked all funny when she said it … oh no, oh no, I bet she’d been Imperiused, and I didn’t realise!”
- Harry Potter and the Half-Blood Prince, Chapter 12 (Silver and Opals)
If the Imperius Curse is done particularly badly, the effects can be quite visible and it’s clear that the cause is an Imperius Curse, because Scrimgeour could clearly tell that the reason Herbert Chorley was impersonating a duck was because of a badly done Imperius Curse.
“Now, about Herbert Chorley – your Junior Minister,’ he continued. ‘The one who has been entertaining the public by impersonating a duck.’
‘What about him?’ asked the Prime Minister.
‘He has clearly reacted to a poorly performed Imperius Curse,’ said Scrimgeour. ‘It’s addled his brains, but he could still be dangerous.”
- Harry Potter and the Half-Blood Prince, Chapter 1 (The Other Minister)
Therefore, it is possible to notice that someone’s Imperiused if the Imperius Curse isn’t cast well enough.
With the Ministry, people noticed changes.
It doesn’t seem likely that when the Minister was Imperiused, anyone knew because he looked blank. When the Dark Lord took over the Ministry, the Ministry and most people in it changed sufficiently enough that it could be deduced from them that some people were Imperiused. However, it didn’t matter if people (whether in the Order, working in the Ministry, or the general wizarding public) knew that the Minister was Imperiused or not. There’s no way for wizards to remove the Imperius Curse, and the Dark Lord controlled the majority of the Ministry before the Minister was Imperiused, so trying to remove Thicknesse wouldn’t have helped. The Order did know that the current Minister was Imperiused and they still weren’t able to get the Ministry out of the Dark Lord’s control.
“The coup has been smooth and virtually silent,’ said Lupin. ‘The official version of Scrimgeour’s murder is that he resigned; he has been replaced by Pius Thicknesse, who is under the Imperius Curse.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
The majority of those still working in the Ministry when the Dark Lord put Thicknesse under the Imperius Curse as Minister either wouldn’t notice or wouldn’t care since they were also controlled in some way by the Dark Lord themselves. The Dark Lord waited until Scrimgeour was surrounded by people he controlled before killing him and replacing him.
“It is a start,’ said Voldemort. ‘But Thicknesse is only one man. Scrimgeour must be surrounded by our people before I act. One failed attempt on the Minister’s life will set me back a long way.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
Anyone still in the Ministry and not controlled by the Dark Lord would have noticed not only the Minister, but also most everyone else in the Ministry, acting strangely once the Dark Lord took power. They wouldn’t see them looking odd unless they were poorly Imperiused, but they likely would realize that their coworkers had drastically changed behavior. Once the Dark Lord and the Death Eaters had Thicknesse, they used him to work against the other Department heads, since he was high-ranking and had contact with the other high-ranking officials they wanted to take control of. Anyone still not in his control then would be heavily outnumbered and see the majority of the Ministry drastically change around them. They wouldn’t necessarily know who was Imperiused, but they’d know things were very different. Since they’d be so outnumbered, they’d also be unable to do anything to change it.
“Yes – my Lord, that is true – but you know, as Head of the Department of Magical Law Enforcement, Thicknesse has regular contact not only with the Minister himself, but also with the Heads of all the other Ministry departments. It will, I think, be easy, now that we have such a high-ranking official under our control, to subjugate the others, and then they can all work together to bring Scrimgeour down.”
- Harry Potter and the Deathly Hallows, Chapter 1 (The Dark Lord Ascending)
It’s unclear whether the general wizarding public specifically suspected that the Minister was Imperiused. Many of them figured out that the Dark Lord was likely behind the sudden drastic changes to Ministry policy, but that was because they knew that it was different from normal, not that they looked at the Minister and thought he looked “odd”.
“Naturally many people have deduced what has happened: there has been such a dramatic change in Ministry policy in the last few days, and many are whispering that Voldemort must be behind it. However, that is the point: they whisper. They daren’t confide in each other, not knowing whom to trust; they are scared to speak out, in case their suspicions are true and their families are targeted. Yes, Voldemort is playing a very clever game. Declaring himself might have provoked open rebellion: remaining masked has created confusion, uncertainty and fear.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
When the Dark Lord controlled the Ministry, people realized he did because of the changes he made. The Order knew Thicknesse was Imperiused, but there are many ways they could have known this that are more likely than that they thought he looked odd. However, it didn’t actually matter if anyone realized the Minister was Imperiused. Since the Order wasn’t able to take the Ministry away from the Dark Lord and had no plans to attempt to remove either the Imperius Curse or the Minister, the knowledge that the Minister of Magic was Imperiused didn’t help them actually stop the Dark Lord’s control of the Ministry.