Word for secret knowledge that is not publishable, but known by “everyone on the inside”

A disgraceful secret such as you describe is called a skeleton in the closet (or in the cupboard).

From Dictionary.com:

10a. a family scandal that is concealed to avoid public disgrace 10b. any embarrassing, shameful, or damaging secret

Generally it's used as a complete phrase (skeleton in the closet/cupboard), but the skeletons (secrets) can be referred to without the idiom in the right context. (Who knows what other skeletons the Prime Minister might be hiding?)

World Wide Words traces the origin back to the fact that medical practitioners illegally and secretly keep skeletons hidden away for anatomical research purposes at a time before it was legal to do such research.

The idea that a skeleton was a figurative representation of a secret shame was once thought to be the inspiration of William Makepeace Thackeray, who wrote in an article in Punch in 1845 that “There is a skeleton in every house.”

Advance information regarding the activities of a corporation is referred to as insider information – particularly when that information can be used for financial gain.

The term

inside baseball

is intended to mean things insiders know (like arcane techniques or for exploiting a rarely encountered rule), things that an outsider wouldn't know about or care about.