The phrase to wash one’s dirty linen in public means: to discuss an essentially private matter, especially a dispute or scandal, in public.

This phrase occurs, for example, in The post-Sturgeon SNP is washing its dirty linen in public. That could be the best thing for it, by Lesley Riddoch, published in The Guardian (London and Manchester, England) of Wednesday 15^th March 2023:

The SNP leadership contest, between Kate Forbes, Humza Yousaf and Ash Regan, seems to prove why political parties should not wash their dirty linen in public. But when there’s nowhere else to do it, the public arena becomes the only location in town. And despite the rancour on display, maybe this long postponed outpouring of difference is no bad thing.

I have discovered that to wash one’s dirty linen in public is a loan translation from the French phrase laver son linge sale en public, and that this French phrase originated in a speech that Napoléon Bonaparte ¹ made to the Corps Législatif (Legislative Body) when dismissing it on Saturday 1^st January 1814 ².

¹ Napoléon Bonaparte (1769-1821), Emperor of the French as Napoléon I from 1804 to 1815.
² Therefore, contrary to what is generally claimed (for example by Christine Ammer in The Dictionary of Clichés: A Word Lover’s Guide to 4,000 Overused Phrases and Almost-Pleasing Platitudes (New York: Skyhorse Publishing, 2013)), Napoléon Bonaparte originated the phrase not on his return from Elba in 1815, but before he was exiled to Elba in 1814.

This is the relevant passage from Napoléon Bonaparte’s speech, as transcribed in various publications in 1814—for example in the Journal de Paris, politique, commercial et littéraire (Paris, France) of Monday the 4^th of April of that year:

« En supposant même que j’eusse des torts, vous ne deviez pas me faire des reproches publics : c’est en famille qu’il faut laver son linge sale ; on ne doit pas appeler tout le monde pour le voir laver. »
     translation:
“Supposing even that I was in the wrong, you ought not to have reproved me publicly: it is within the family that one’s dirty linen must be washed; one ought not to call everybody to see it washed.”

Variants of the phrase pronounced by Napoléon Bonaparte include the following:

1– : From De la conduite du Sénat sous Buonaparte, ou les causes de la journée du 31 mars 1814, avec des détails circonstanciés sur cette journée mémorable (Paris: Lebègue, Petit, Blanchard, 1814), by Étienne Le Hodey de Saultchevreuil (1754-1830):

Quand on a du linge sale, il ne convient pas de le laver en public.
     translation:
When one has dirty linen, it is not proper to wash it in public.

2-: From Qu’est-ce que la liberté de la presse, selon l’article VIII de la Charte constitutionnelle ? Ou qu’est-ce que l’existence physique, morale et politique ? (Paris: Chez Chanson, 1814), by Soulety:

[Buonaparte dit] que l’on devait laver son linge sale en famille, et non pas en public.
     translation:
[Buonaparte said] that one must wash one’s dirty linen within the family, and not in public.

3– : From Examen des Observations publiées à Paris le 4 avril 1815, sur la Déclaration du Congrès de Vienne, du 13 mars, published in Le Journal Universel (Ghent, Belgium) of Tuesday 18^th April 1815:

Je vous apprendrai à savoir du moins laver notre linge sale en famille.
     translation:
I will teach you to know at least how to wash our dirty linen within the family.

The earliest occurrences of the phrase to wash one’s dirty linen in public that I have found are as follows, in chronological order—all the texts specify that Napoléon Bonaparte originated the phrase:

1-: From A Defence of the People, in reply to Lord Erskine’s “Two Defences of the Whigs.” (London: Published by Robert Stodart, 1819), by the British politician John Cam Hobhouse (1786-1869):

We, radical Reformers, will not, as Bonaparte said, wash our dirty linen in public; so we will not regret the past, but hope for the future.

2-: From Sketches of Society. Paul Pry on his Travels.—Letter IX., published in The Literary Gazette, and Journal of Belles Lettres, Arts, Sciences, &c. (London, England) of Saturday 3^rd June 1826—in the following, Paul Pry is talking with “a French savant” in Paris:

“Will you tell me why the French Academy, the Academy of Inscriptions and Belles Lettres, and the Academy of the Fine Arts, close their doors against all strangers? I understand it was not always so.” “Mr. Pry, you have put a puzzling question; yet I will strive to answer you. The Academy of Sciences admits strangers, because the objects under discussion generally admit of demonstrative proof, whereas, in the other Academies, the imagination holds an equal empire with reason and common sense, and sometimes even triumphs over them; hence it frequently happens that the old proverb is realised of, many men, many minds; the discussion waxes warm, and they recollect the expression of Buonaparte, that none ought to wash their dirty linen in public.” “I don’t understand you, for I see the women wash linen in public daily in the Seine.” “My dear Mr. Pry, that is not what the Emperor meant; he wished to impress on the parties, that family quarrels ought never to be heard but by the members of the family, in order to avoid scandal.”

3-: From the transcript of a speech that Douglas Kinnaird made in defence of the Marquis of Hastings at the General Court of Proprietors of the East India Company, held on Wednesday 26^th September 1827—published in The Morning Herald (London, England) of Thursday 27^th September 1827:

He regretted to hear any comparison instituted between the merits of the Noble Marquis and those who preceded him as Governors of India. There was no necessity for the contrast. The fame of either did not require it. He hoped the Meeting would avoid those annoyances, and recollect the advice of Bonaparte—“Not to wash the dirty linen in public.”