Someone who intentionally says false statements and/or facts about someone else (directly related to and/or about them) with whom that person is in a conversation (with or without other people around), in order to embarress that person, bring them down or prevent/break the peace.
You can use rumormonger, rumor-monger, or rumor monger to describe such a person, with the assumption that rumors usually are false, or at least unverified.
a person given to spreading rumors, often maliciously.
rumor: a story or statement in general circulation without confirmation or certainty as to facts
If you want a more negative connotation, you can use slanderer.
slander: a malicious, false, and defamatory statement or report: a slander against his good name. Related forms: slan·der·er, noun
Consider, badmouth [someone]
To criticize or disparage, often spitefully or unfairly: those cross-Atlantic aficionados who persistently idolize the British theater and bad-mouth Broadway (Benedict Nightingale). American Heritage® Dictionary of the English Language, Fifth Edition