They're Called "Personal Issues" for a Reason

"Everyone has something at home they don't want anyone to see; that is one of the functions of a home, to provide a spot to keep such things."

—Nero Wolfe, The Red Box

Wear your broken arm inside your sleeve.

—Chinese proverb

Keeping Secrets Sucks. We all know that. However, sometimes, the only thing that sucks more is telling those Dark Secrets. Not as common as some other Stock Aesops due to its status as a somewhat Family Unfriendly one. It serves as either a subversion or aversion of the common belief that friends don't keep secrets from friends. The idea behind this is that everybody has certain things that they'd really rather not have to say, and it isn't right to force people to say those things just to satiate one's curiosity. Say The Atoner wants to start over after his Dark and Troubled Past, for example. Or say someone doesn't want anyone to find out about a tragedy that affected them previously and offer undesired sympathy. On the other hand, this usually does come with an exemption for if there's something they're keeping secret that affects you directly.

It can also be used as a handwave in order to keep The Masquerade going. Basically, a stranger comes to town and ingratiates himself in a group. Usually, this will be a main character. In order to explain why no one in the group is curious about just why this person Walking the Earth has just wandered into town, this aesop is invoked. Sometimes, this can also be the explanation for an Unusually Uninteresting Sight. See also There Are No Therapists.

Examples of They're Called "Personal Issues" for a Reason include:

Anime and Manga

Higurashi no Naku Koro ni uses this Aesop heavily, particularly in Onikakushi-hen, where it serves as a good chunk of the solution, and Tsumihoroboshi-hen. Justified as Keiichi had just moved into Hinamizawa, and didn't know or realize just how uncomfortable the village was with said secrets.
- In Tsumihoroboshi-hen, this trope is also explicity pointed out. When Rena reveals to Keiichi that she knows he was the perpetrator of some airsoft gun shootings in his old town, he breaks down and confesses to the others. They react with confusion, not understanding why he's telling them this. When Keiichi says that Friends Don't Keep Secrets From Friends, Mion flat-out states that if having friends means confessing every single wrong you've ever done (particularly ones that don't matter anymore), then she doesn't want any friends. That being said, the story also goes out of its way to point out that if you find yourself in a situation that you can't handle alone, then it's okay to rely on your friends. This particular scene is also important in that his friends' kindness toward Keiichi triggers his memories of Onikakushi-hen. Namely, what really happened when he killed Rena and Mion.
Kaoru says something along these lines at the beginning of Rurouni Kenshin when she invites Kenshin to live at the dojo with her, basically saying that what happened in his past is none of her business.
In Ghost in the Shell: Stand Alone Complex, Paz's Day in The Limelight episode revolves around his habit of sleeping around and a girl he wanted to stay with but couldn't or so he told her. He basically tells everyone it's a personal matter (when he realises what's going on) and leaves behind an address so they can clean up the aftermath;

"It's a personal problem. I'll deal with it myself. If I become an embarrassment to Section 9 you can cut me off."

Cowboy Bebop. Everyone on the Bebop has issues (except Ein). Everyone knows that all the other people have issues too, but not what caused them. And none of them ever pry because Jet and Spike both believe heavily in this trope, Faye and Ed don't care, and Ein's a dog so it's not like he can just play therapist. The overall arc of the show is seeing how their old deeds come back to haunt them and how this slowly, gradually, makes them open up to each other.
At the end of Eden of the East, after Saki successfully finds Takizawa's mother, who abandoned him when he was five, she vanishes without a trace. Saki brushes this off with "I guess she has reasons she can't tell anyone."
Averted in Saiyuki, surprisingly. At first glance it seems like this would be the case, since all four main characters don't always get along and all have some extremely fucked up issues they definitely keep from everyday conversation. A couple situations help subvert it though. The first is that Goku HATES people keeping secrets from him, as seen when Gojyo's backstory is revealed (though it is Hakkai that tells that particular tale, Gojyo isn't the type to mind Goku knowing). The other is that the one who's most touchy about his issues, Hakkai, and who tends to bottle things up had all the others playing active roles in his backstory. In a profile of Hakkai Minekura did states that Hakkai desperately wishes he could employ this trope sometimes.

Literature

Callahan's Crosstime Saloon: Callahan's has a strict "no snooping" rule about the guests' personal issues, which is enforced by the pianist's blackjack. If anyone wants to share their problems (whether it's about their messy divorce or the alien armada poised to annihilate Earth) they are welcome and even invited to do so, but get pushy and you'll end up face down on the driveway with a nasty welt on your head.
In The Inheritance Cycle, Murtagh was fully prepared to spend his entire stay with the Varden in "some rat hole chewing on hardtack" due to his refusal to allow them to read his memories. Although he had confided in Eragon beforehand, he didn't want the knowledge to spread that he was the son of Morzan, the Rider who helped bring Big Bad Galbatorix to power.

Live-Action TV

Was sort of the aesop of a How I Met Your Mother episode where Robin didn't want the group to know why she hates malls. Her secret eventually came out anyway thanks to Barney dredging it up, but she gets over it pretty quickly. So... the episode was about this but in the end there wasn't that much a point made either way because Robin ultimately didn't seem to mind that much that everyone found out.
In the third season of Leverage, it turns out that Eliot used to work for the Big Bad that they've been chasing for most of the season and has not seen fit to enlighten the rest of his crew. He's not willing to elaborate on the details.

Eliot: Don't ask me that, Parker. Because if you ask me, I'm gonna tell you. So please, don't ask me.

Morgan from Criminal Minds is willing to risk jail time and the destruction of his career rather than admit to his teammates that he was sexually abused by the UnSub of the week. They figure it out anyway.
- Lampshaded by Gideon:

Hotch: There are larger implications. I can't have someone on this team who keeps secrets.
Gideon: Come on, Hotch, we all have secrets. Would you want us profiling you?

Mad Men: Joan forbids the Sterling Cooper secretaries from crying in the break room because she believes that personal issues are meant to be left at home.

Video Games

358/2 Days uses this as a handwave in Olympus Coliseum after Phil finds out that Roxas wasn't, in fact, referred to him by Hercules.
Squall in Final Fantasy VIII takes this attitude for a good half of the game, believing that everyone has to deal with their own personal issues on their own and there's no point in discussing them. The rest of the cast disagrees.
The concept is paraphrased by Medoute at the beginning of Blaze Union's B route when she's trying to dissuade Siskier from butting into Aegina's past. (Of course, Medoute has things that happened in her past that she doesn't want to talk about, either.) Since Aegina's personal issues start causing huge problems for the party later on, though, she winds up having to discuss them and the point becomes moot. It's also suggested that if the other characters had known a little more about Garlot's personal issues, he might not have taken such an emotional beating over the course of the canon route.

Web Comics

The Inexplicable Adventures of Bob. When Fructose Riboflavin is at a very low point, alone in a dark room, he starts muttering to himself about the childhood tragedy that started him on his road to villainy. When he realizes Bob overheard him, Riboflavin doesn't take it well. He takes it even less well when Bob points out that said tragedy doesn't justify any of his crimes.

Western Animation

In the Avatar: The Last Airbender episode "Zuko Alone", the farmer who takes in Zuko says that his past is his own business. True enough, the revelation to the whole village that Zuko is the Fire Nation prince makes them all turn against him.