Hmm! Quite the suave expression.<2020 “Walsh, who unsuccessfully challenged Trump for the 2020 Republican nomination, said: “I can tell you, publicly and privately conversations I’ve had, they’re scared shitless, all of them, any Senate Republican on the ballot.”—theguardian.com, 11 July>

I’ve always wondered what the literal meaning of ‘scared shitless’ implies. On the face of it the ‘shit + -less’ would seem to imply that the fear prevents one from shitting. But when people are very, very scared they are known to literally “shit in their pants.” So it seems that although ‘scared shitless’ could mean that you are so scared you can’t shit, that clashes with reality. So, the alternative is that 'Fear has caused one to shit so much that there is nothing left – one can shit no more!' Wow! That's heavy.

Now for what the pros have to say on this delicate subject:
The Oxford English Dictionary
shitless adverb (coarse slang): To an extreme extent (especially of mental states). Usually in phrase scared shitless.
Origin: Formed within English, by derivation. Etymons: SHIT noun, -LESS suffix.
Etymology: <SHIT noun + -LESS suffix, in allusion to the loss of bowel control that can accompany extreme fright.
______________________________
Well, “in allusion to the loss of bowel control,” can’t be denied. But to be more precise, I would go with my above more accurate and elegant alternative.
______________________________
Since the OED doesn't actually provide a definition for scared shitless, to be complete, I looked elsewhere:
IdiomsThefreeDictionary.com
scared shitless rude slang: completely terrified <I was scared shitless when I woke up to the sounds of an intruder in my house.>
______________________________
The following quotes are from The Oxford English Dictionary and archived sources:
____________________________<1924 “Vulgar figures of speech often embody truths expressed with dynamic accuracy... ‘I knocked the shit out of him.’ ‘I will scare him shitless.’”—G. Stragnell in Psychoanalytic Review, 11 page 320>
<1957 “But they got this Union, it's not healthy to buck them and The Minaret is scared shitless to run an editorial on these foul abuses.”—Letters of W. S. Burroughs (edited by Oliver C. Harris), 28 January (1993), page 352>
<1976 “The self-appointed custodians of public morality who campaign against pornography because they're simply scared shitless by it.”—New Musical Express, 12 February, page 24/1>
<1996 “Everyone except Buffy (in hot dress, out of breath) is scared shitless, giving the monster a wide berth.”—Buffy the Vampire Slayer: Script Book (2000) 1st Season, I. page 201 (stage direction) by D. Greenwalt>
<2002 “Whatever you're doing, you scared my boss shitless.”—Cleaning Up (TV shooting script) by G.P. Pelecanos, 38 in Wire (O.E.D. Archive)>
<2012 “Yup, you [[Democrat, Rob Zerban]] should be scared shitless of me [[Republican, Paul Ryan]], because guess who isn’t? The people of Wisconsin. They love me. Republicans and Democrats there love me.”—Chicago Tribune (Chicago, Illinois), 23 August, page TAB-8> [[election of U.S. House of Representative seat for Wisconsin]]
<2019 “The cynic . . . recognizes that the U.S. space program was essentially the political response of the cold war threat posed by the Soviet Union. ‘People were scared shitless of the Reds taking over.’”—The Ottowa Citiizen (Ottowa, Ontario, Canada), 20 July, page D3>
Ken Greenwald – July 14, 2020