I can only speak for myself but when I was in the military my job required a Top Secret SI security clearance. When my enlistment was ending I was approached by the major govt. agencies, CIA/NSA as well as some private contractors to recruit me. So that's one way. Another is off college campuses.

A lot of the "every day" jobs like translator, data analyst, etc. can be applied to directly. The government (in the US at least) even has websites for this:

https://www.intelligencecareers.gov/

https://intelligencecareers.usajobs.gov/Search/Results?cmco=IC%20Intel&p=1

They also sponsor events to look for talent sometimes, like the "NSA Codebreaker Challenge" which tests IT and programming skills. The winners of these challenges may be recruited.

Ads(literally on instagram), posts on the main gov hiring website, campus career days/events, and generally name recognition.
Depends how senior of staff but most promote internally. Director levels are appointed.

Spooky agencies who do Intel things do reach out to promising candidates, high language skills, knowledge of necessary areas, specialized expertise like certain ex-mil jobs. A common theme was to have agency contacts at schools and have them point out promising people to recruiters.

It depends on skill set. And what's needed at any of the "alphabet soup" agencies.

Most require degrees.

If say, you have degrees in Pashtun and Forensic accounting, while dabbling in pre law and mandarin.

They'll find you.

Anything else you go to USAJOBS.gov and apply.

[removed]

[removed]

[removed]

Pretty much the same as any large company honestly. Recruiters, job fairs, online applications, etc. To actually start work however, you will typically need to get a security clearance if you don’t already have one, and that’s a long and involved process. Much more so than what you typically think of as a background check.