Toph sees with her Earthbending. Then, accidentally, despite being traditionally blind, she activates Eagle Vision. Fortunately for her, she does this while in the presence of Xin Fu. The guy may be generally an ass, but he's also an Assassin Mentor. The Earth Rumbles are an undercover/secret-within-a-secret sort of training arena and method for discovering potential Assassins.
Her father is a Templar, and her mother is completely ignorant. Toph does her best to spy on her father for the Assassins, but not being able to see in the traditional sense does hamper her efforts a bit there. She also keeps her head down around her father as much as she can because of the Assassin/Templar thing, and not just because she has to pretend to be his helpless, fragile, obedient, blind little daughter.
Then Aang comes along, asking her to be his Earthbending Teacher. Toph personally arranges for the 'kidnapping' by Xin Fu - he's happy to go along with it, since it will get a nice little bundle of money out of the Beifong/Templar coffers and into the hands of the Brotherhood. She talks it over with the grumpy Mentor, and a conclusion is reached.
If Toph can use the 'kidnapping', and her 'rescue' of the Avatar, to infiltrate the Templars through her father, good. If not, she will assassinate him, and then leave, joining the Avatar to be his Earthbending Teacher. Toph is definitely upset about it, that her first assassination is her controlling - but caring - father, but she still does it.
Zuko was raised in a Templar environment, but his mother (who was forced to marry Ozai because Fire Nation Politics), did her best to raise him as an Assassin when her husband and his family weren't looking. When she was forced to flee the capital, she left a small chest under Zuko's bed which contained letters, diaries, and weapons.
Zuko walked the razor's edge alone since then, studying the ways of Assassins and Templars alike, the former in secret. When he faced his father in the Agni Kai, he was genuinely horrified, upset, and even more frightened. He wasn't anywhere near ready to face someone as powerful and as potent as his father. Not like this. Maybe as an Assassin he might manage to kill his father, but in an Agni Kai? Not possible. Never mind that, despite everything, Zuko still couldn't help but love his father, even when he knew that as an Assasin, he shouldn't care for the Templar Grandmaster that was also the Fire Lord. But faced with banishment? It was a relief.
Until Iroh invited himself along. Iroh, who was also a Templar. Except, as the days and weeks and months went by, Zuko got the idea that Iroh had left the Templar ideals behind for something else. He wasn't an Assassin though, that was for certain. So Zuko didn't dare to reveal himself as an Assassin to his uncle, just in case. His Assassin training was still done in secret, and he balanced on a new, sharper, more complicated razor as he listened to his uncle.
His uncle who had been a Templar, would never be - and probably never trust - an Assassin, and who was something else. Zuko hadn't known there was a potential third side to the conflict. He wasn't altogether sure that he liked that there was. He wished that he had a Mentor that he could contact for advice, that he was in touch with any branch of the Brotherhood. All he had though, was the codex, diaries, weapons, gear and manuals that his mother had left in a chest under his bed. The night she disappeared.
He missed her.
Sokka and Katara are both completely ignorant of the Assassins and the Templars, but if the two secret societies had been described to the Water Tribe siblings, they wouldn't have been at all surprised to learn that the Fire Lord was a Templar. They never do find out.
Avatars of the past were Templars about half the time, completely ignorant of the struggle 49% of the time, and are Assassins that remaining 1%. The Avatar Spirit forbids and prevents the previous Avatars from coming out and telling the living incarnation about the Assassin/Templar thing, at all, so none of them have any idea about their previous lives stance on the matter until they're dead and have joined with all the previous lives.
The proportion of lives where the Avatar was a Templar is in part because of the Templar belief that, for there to be peace in the world, the world needs for there to be someone in charge, monitoring and guiding that peace. Someone like the Templars in general, or the Avatar in specific - provided that the Avatar is a Templar.
Aang, having been only twelve when he ran away from the Air Temple and got himself frozen alive in the ice of the South Pole, was essentially ignorant of the struggle. Air Benders, in general, were neither Templars or Assassins, believing instead in finding peace through detachment from the world. His own interpretation of his Destiny as the Avatar is quite Templar-ish at times though, and worries Zuko and Toph on the quiet.
Even if they do both agree that Fire Lord Ozai does need to be removed, it's the question of what will come after that worries the two Assassins. Not that either of them realise that the other is an Assassin. Not at first, anyway. Not until Zuko has been part of the Gaang for a couple of weeks, and Toph finds him doing maintenance on his hidden blade. An item that he hasn't had out, and certainly hasn't worn, since he first appeared with the offer to teach Aang Firebending.
Blades that are attached to a forearm bracer, and meant to be hidden under a sleeve, don't really go well with short sleeves. Toph, who hasn't worn hers since she assassinated her father before joining the Avatar (and for the same reasons - short sleeves), goes to get her bag and returns to Zuko's side without him ever knowing he'd been spotted at the task.
Until she sits down next to him and asks "Can you check for me that I got all the blood out?", and holds out her own bracer with its hidden blade.
"Nothing is true," Toph recited softly.
"Everything is permitted," Zuko answered, and took the blade. "You did pretty good at getting it clean, actually," he offered, even as he went over it.
"Thanks. I was going by the feel of it, since I can't see the blood stains and all."
"Are the others -?" Zuko questioned, half-hopefully.
Toph shook her head. "Not a clue," she denied with a huff. "And by that I mean, they don't have a clue."
Zuko nods in silent acceptance of this. No, he doesn't get flustered with himself for giving a non-verbal answer to the blind girl. He's been around her enough by now to figure out that, as long as they're on solid ground, and they are right now, she can 'see' him just fine.
When Suki joins them, Toph and Zuko both wonder, but the Kyoshi Warriors are another group altogether. Kyoshi had been one of those Avatars that had lived in ignorance of the Templar/Assassin stand-off. She'd had her own, quite loud, opinions of how things should be done though. Apparently, she'd really messed up a lot of plans during her life time. For both the Assassins and the Templars.