I'm looking for a specific verb that mean 'giving people shit' (as in teasing them, keeping them honest).
It needs to capture that the teasing is warranted, and that the criticism is correct.
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Consider razz. It is an informal verb but captures both teasing and criticism.
to make playful or unkind comments about (someone) [MW]
to deride, jeer; to mock or make fun of (a person or thing). [OED]
To poke fun at someone, but benevolently, is to "josh"
intransitive verb: to engage in banter: joke
transitive verb: to tease good-naturedly: kid
Examples of JOSH
• “Don't take him seriously. He's just joshing”
• “Don’t get all hot and bothered! I'm just joshing you”
Synonyms: chaff, jive, joke, tease, kid, rally, razz, rib, ride, roast
Apart from the plethora of UK colloquial slang, the only single verb I can think of that I still see regularly in books or hear in conversation is;
to criticize and laugh at (someone or something) for being bad, worthless, or unimportant
to laugh at or make fun of (someone or something) especially by copying an action or a way of behaving or speaking
"They continue to mock the idea of a new government"
For something a little more light hearted;
something said or done to cause laughter
an utterance (as a jeer or quip) intended to be taken as mockery or humour
"you should know that our teasing was done entirely in jest"
The word is chastise:
to criticize (someone) harshly for doing something wrong
- to inflict suffering upon for purposes of moral improvement; to discipline, especially by corporal punishment.
- to criticize severely.
- Archaic. to restrain; chasten.
- Archaic. to refine; purify.
You could intensify it to chasten:
to correct by punishment or suffering : discipline; also : purify to cause to be more humble or restrained :
to inflict suffering upon for purposes of moral improvement;
If the connotation of moral improvement, wasn't essential, I would go with harass or harangue.
The term twit might be useful
Tease or taunt (someone), especially in a good-humored way.
Oxford Dictionaries Online
It is usually used when the object of the poke is in the wrong
To taunt, ridicule, or tease, especially for embarrassing mistakes or faults
To make fun of; tease.
Supplement: Also consider chide
Scold or rebuke: she chided him for not replying to her letters
[WITH DIRECT SPEECH]: “You mustn’t speak like that,” she chided gently
Oxford Dictionaries Online
The (probably AAVE) word I always heard for this growing up was
ragging. It was kind of a sport at my school. You never knew when an impromptu game of Dozens might break out, so it paid to keep your skills sharp and figure out everyone's weaknesses up front.
Ragging doesn't have to be true. However, it will be far more effective the more truth it contains. For instance, someone with a skinny mother is probably not going to be impacted very much by a "Yo' momma so fat..." snap.