Generally, my group is very open about their characters.
In my group/campaigns, each player has always known what each other PC's race and class are/were. There's always free discussion when they level about abilities, feats and using skill points, so there's a general knowledge about what other PC's can do and possibly their strengths and weakness. Everyone quickly forgets what other PC's have for Ability Scores, actual skills (ranks/proficiencies, etc), Spells, or whatever...
When I'm a player myself, I like to keep some details about my character quiet (maybe not a secret, but I won't advertise it) either as part of my background, or just be a little surprising occasionally. A few of my players are like that too.