(PW)
discuss personal problems openly
He is a very private person. If he has a problem in his family he doesn’t want to air his dirty laundry in public.
(MW)
(UsingEnglish.com)
If you air your dirty laundry in public, you reveal aspects of your private life that should really remain private, by telling a secret, arguing in public, etc.
(thePhraseFinder)
"Air your dirty laundry (or linen) . . ." means to tell scandalous or unflattering stories about yourself, to reveal things from your private life that people usually don't want others to know. Sometimes the phrase is used when the privacy that's invaded is the privacy of a couple, not an individual. For example, a husband might say to friends "Mary and I got so mad at each other last Friday that we didn't speak all weekend," at which his wife says "John, let's not air our dirty laundry in public." It's a metaphor. People would normally find it embarrassing if others saw their clothes and sheets and towels that need washing, and if they're not embarrassed, then the witnesses will be. The same goes for making inappropriate confessions and revealing unpleasant family secrets.
"The amount of women in London who flirt with their own husbands is perfectly scandalous. It looks so bad. It is simply washing one's clean linen in public." Oscar Wilde.