They shouldn't do it. They knew well not to read their fellow champion's journal. But if it is laying so out in the open, it is as if Link invited them to read it. Zelda sure read it as such.
"Let's see. Ah, he wrote about Daruk!", said Zelda, flipping the pages. "He wrote, 'I admittedly have not much to say about the Goron, just that he is great fun to be around. He is probably my second favorite of the champions.' Huh, who is number one?"
"Second?", asked Daruk. "I assume Mipha is on top."
"Or Zelda.", guessed Mipha, "Link is her personal knight after all... where is he anyway?"
"His family requested that he visits them.", explained Zelda. "He didn't look excited."
"What did he write about me?", asked Urbosa.
"Urbosa! Well, he wrote, 'Urbosa is so utterly strong. It is impressive how she can fight in heels, considering how often Zelda and my sister complain about them. I wonder if Urbosa would be more dangerous without them?' We need to test that.", read the princess of Hyrule. "He also wrote, 'Either way, Urbosa is scary, definitely not someone whose bad side you want to be on.' He is afraid of you?"
"I won't say he is afraid of me.", stated Urbosa.
"What about me?"
"Okay, Revali. He wrote about you, 'Despite this Rito being the third most arrogant person I ever met, I have to admit he is probably the best archer. I would like to ask him to teach me archery, but I doubt he would agree.' Oh, Revali?"
"Fine, I do it.", agreed the Rito. "But only because you said so, your highness."
"I wonder who he thinks are more arrogant than Revali?", wondered Mipha.
"Let's see... he wrote about you... oh my, by Hylia! 'I, Link, love Princess Mipha of the Zora... and it is very painful to do so.' Oh my...", read Zelda. "Mipha?"
"We... have been seeing each other for a while.", answered the other princess. I have no idea what he thinks is painful about it. Please continue."
"Of course. 'First of all, we're different species. While not banned, I do not want to ruin Mipha's reputation. Secondly, albeit in the same vein, different status in society.' Oh, he worries about that?", said Zelda. "Oh, Link. 'Not to mention, I am bound to Zelda. Can I even be with anyone without it conflicting with my duties? It hurts but I hope Mipha breaks up with me. I just can't.' Oh, Mipha."
"I need talk with Link.", said Mipha.
"To be fair,", chimed Urbosa in, "His duties to Zelda really complicate relationships."
"Let's see what he says about me. 'I'm probably get killed if anyone reads this, but I think I hate Princess Zelda.' Huh?! Why? Why does he think he would be killed?" Zelda didn't understand.
"Well, you were quite rude to him at first.", chimed Urbosa in. "And your mother was quite tyrannical to the knights..."
"I suppose. 'Does the princess want to die? I may be her knight, but that doesn't mean that she just can throw herself into danger. Zelda makes me want to die, let a different knight fail king Roam.' I-I..." Zelda can't believe what she reads. "Does he really think this about me?"
"Well, it is his journal.", said Revali. "I wonder what he thinks of us when he discovers we read his journal."
"Oh... Oh! After that he seems to write about his family!"
Link's family is an odd case. They are well alive. But Link visits them rarely. In fact, Link seemed to suffer when his family requested he visits them.
"My father and I met them a few times.", explained Zelda. "He has a sister and a brother, both younger. They're twins, I think? I just know Link was awkward around them and acted rather cold, especially with his father."
"Was his father a knight?", asked Mipha.
"No, but his ancestors. Link's father is a trader, works for his wife's families company.", said Zelda. "I assumed he would pretend they don't exist in this, considering how he behaves around them."
"Really?", asked Mipha. "Well, he seemed to avoid the topic of his family."
"Let's see. 'On one hand I am happy to be free from them, now they can focus on their favorite child without me being a bother. But I didn't meant to slave away to that incompetent princess. I hope I am dead when she becomes queen. The day King Roam dies is when Hyrule falls. Even I can't stop that.' Oh..."
"Oh, the favorite child thing or what he thinks of you?", asked Daruk.
"Both? 'I can't decide who is more arrogant, the princess or my brother?' I need to fix this."
"I disagree. He does his duties despite his feelings.", said Urbosa. "Knowing your father, we better keep silent about this. Especially you, Revali."
"Fine, I'll train him in Archery anyway.", said Revali.
A century passed. Link defeated the Calamity. Zelda was in Link's house in Hateno. They were working on rebuilding the kingdom.
Right now Zelda was reading an old journal. One reminding her of a certain shame.
"You were right, the day my father perishes is the day Hyrule falls, but you were able to stop that.", said Zelda.
"Uh, what are you talking about, Princess Zelda?", replied Link. He is still amnesiac. The memories he gained back apparently didn't include the journal, at least not the section she and the champions read shortly before Hyrule was overrun. "And may I ask what you read there?"
"Just your journal from a century ago.", replied Zelda sheepishly. She saw Link becoming pale with horror. "Relax, the champions and I read it a century ago already. Well, parts of it."
"That makes it worse.", said the hero of Hyrule.
"Anyway, you wrote on this pages-" She showed him the pages in question. "-that Hyrule will fall when my father dies. Looks like you were right.", explained Zelda. "But you were wrong about one thing, you did stop the Calamity."
"But with latency.", said Link. He skimmed the journal. "Oh, Koroks. I-I am very sorry to have written about you like that."
"No need. In some aspects, you were right.", said Zelda. "Seriously, it is okay."
Link wasn't that sure about it. He was just thinking, 'Why did my old self want to be a knight? You have like zero privacy.'