Repeating a rumor that you started? I've heard that repeating a rumor by denying it only serves to reinforce the rumor. 

https://www.linkedin.com/pulse/20141110142443-1967426-how-firms-should-fight-rumors/

Try to avoid reinforcing minor rumors by repeating them in order to deny them.

https://www.psychologytoday.com/us/articles/200811/the-8-laws-rumor-spread

Even a denial can be a repetition of a rumor." (Just ask Senator John Kerry, whose 2004 presidential bid sunk thanks to whispers about his swift-boat service in Vietnam—even though most of the media stories were about how the rumors were false.)

https://www.wikihow.com/Stop-Rumors

If it was you who started the rumor in the first place, don't deny it. Instead of reacting to what others will think of you, admit what you did wrong out of character.

But what about in cases where I told something about myself that was less than flattering, or even downright embarrassing, and am now regretting having said? Does the same principle -- repeat it in any way by denying, asking to keep silent about the matter etc... -- result in heightening the possibility that it will spread to other people, because of the effects of priming?

Or is it the case that if I ask him/her specifically to not repeat the rumor, that he/she will more likely comply compared to if I kept silent about the matter?
The Truth Effect:


  Repetition makes statements easier to process relative to new,
  unrepeated, statements, leading people to believe that the repeated
  conclusion is more truthful. ... The illusory truth effect plays a
  significant role in such fields as election campaigns, advertising,
  news media, and political propaganda.


A seminal experiment is Begg et al (1992), in which subjects were presented with statements that they were told were false, but later rated them as truer than statements they were not previously exposed to.  Merely being exposed to statements makes them more familiar, and hence judged more true, and even being told that they are false does not counteract this effect of familiarity.

More recent studies have replicated these results with fake news stories.  For example, Polage (2012):


  ... repeating false claims will not only increase their believability
  but may also result in source monitoring errors.


And Pennycook et al (2017):


  ... even a single exposure increases subsequent perceptions of
  accuracy ... Moreover, this “illusory truth effect” for fake news
  headlines occurs despite a low level of overall believability, and
  even when the stories are labeled as contested by fact-checkers or are
  inconsistent with the reader’s political ideology. These results
  suggest ... that tagging such stories as disputed is not an effective
  solution to this problem.


This is consistent with belief perseverance.  The broader familiarity effect that causes illusion of truth is called processing fluency or cognitive ease.