Though the rules are undetermined, I don't think sharing character sheets themselves is beneficial in the long run. Though I agree some information should be shared, as it's obvious for the characters to know (such as race, gender, appearance), there are a few problems I can see with fully disclosing your character to the other players:
Page 2 is quite personal.
There's your whole backstory, your quips and qualms, and your most prized possessions listed there in black and white. Though it's important for that character's player to know, the only time you'd ever know another person that well is if you've spent a lot of time with them. So if your party are childhood friends, then they should already know that information, but if not, I feel that would be as violating to the character as if I would ask you all sorts of personal information just because you asked a question on this site...
That said, the one person who should know your character's back story besides you is your DM. Since they are forming the story to challenge your character and can award points based on how well you play your character, it would make sense for them to know who they're dealing with.
A character might be shrouded in mystery.
There are many reasons a character can be mysterious. Be it their profession, their past experience, their trust towards strangers, or their backgrounds they want hidden (like Criminals or Urchins wanting to change their ways), as a character, they would naturally be reluctant to share personal info with people that they just met. Those kinds of revelations are often earned by building trust between characters, an act of role-playing that your players can involve themselves in, immersing them further into the story. In my experience, if a player already knows a character's mysterious past, they are less likely to have their own character pry for that information as well.
Doing so might feel like hand-holding.
I might be wrong, but D&D has always been, for me, a way to express myself in new and fun ways. Since every action I make has irreversible consequences, it allows me to fully engage in playing the character and exploring a new aspect of myself. In that way, if my DM was to tell me "You don't want to go into the forest, there's an ambush waiting" or "If you do that, you'll get robbed", it makes the game feel less personal and more like I'm just a pawn in the DM's fantasy. Likewise, if I was invested in a character as I made them, and I was forced to expose their secrets to others, I'd feel like I'm exposing myself to them as well, which would break the fun of the game.
Again, the rule is that it's up to the DM and the players in the end, and some people might enjoy that full exposure, but I found that, for me, the best method of introducing my character beyond what I look like and what first impression I give, would be to leave me do it my own way, to grant me full control over my character from the start. And if I want to know about another character, as a player, I can simply state that I ask them.