(Judaism refers to “gossip” as one kind of “loshon ha-ra” — literally, “evil tongue.” The rabbis consider it tantamount to murder, because “the blood drains from the face” of the victim of gossip.
I heard the following story about St. Philip Neri from Fr. Michael Aparo. Looking it up online, I found that there are several versions of the story that differ in details but not in the essential point. The narrative sub-theme of confession/penance might seem like something foreign to Judaism but in fact, many Orthodox and Hasidic Jews consult their rabbi or rebbe about how atoning for a particular sin.
In this story, the “sin” is actually the behavioral or character failing of gossiping, which is certainly something that one does more than once. “Loshon ha-ra,” often translated as “gossip,” actually refers to any number of ways that speech can be misused.
“Penance” for a repeated “sin” would therefore fall partly within the province of what Judaism calls “Musar” — character improvement.
But the point about the virtual impossibility of making restitution for gossip, is well taken.
The way that the woman seeks the aid of a holy man to conquer her fault is very reminiscent of Hasidic stories about visiting your rebbe or another tzaddik for similar help.
This anecdote has a quality very much like the aggadah of the Talmud.)
“He [or she] who robs another of his [or her] good name can never repair the terrible injury he [or she] has caused.
A woman once came to St. Philip Neri, (1515-1595) confessing that she had a terrible weakness for gossiping…St. Philip then told her, ‘Do this for your penance, my daughter. Go to the market and buy a dead chicken that still has its feathers, and carry it through the streets. As you go along, pluck out the feathers one by one and scatter them on the way. When you have done this, come back to me.’
The woman fulfilled her strange penance, and returned to him. Naturally, she was very curious as to what he would say next. The Saint praised her obedience in following his first command. But now he told her to go again through the streets, and gather all the feathers she had plucked out! She was surprised by this command, and answered him, ‘Father, that is impossible. For the wind will have carried them away in all directions.’
And so, St. Philip Neri told her: ‘Just as the wind disperses feathers in all directions, so it is with gossip. The persons who hear it go their separate ways and carry it far and wide, so that it is beyond reach or correction’.” 
I found essentially the same “aggadah” or “ma’aseh” (story) in an online Jewish discussion of loshon (or lashon) ha-ra, in which a rabbi gives the same direction as did St. Philip Neri. Note that the author of this online discussion, who otherwise scrupulously cites his sources, cites none for this story, other than to call it “Chasidic.”
“The harm done by speech is even worse than the harm done by stealing or by cheating someone financially, because amends can be made for monetary harms, but the harm done by speech can never be repaired. For this reason, some sources indicate that there is no forgiveness for lashon ha-ra (disparaging speech). A Chasidic tale illustrates this point: A man went about the community telling malicious lies about the rabbi. Later, he realized the wrong he had done, and began to feel remorse. He went to the rabbi and begged his forgiveness, saying he would do anything he could to make amends. The rabbi told the man, “Take a feather pillow, cut it open, and scatter the feathers to the winds.” The man thought this was a strange request, but it was a simple enough task, and he did it gladly. When he returned to tell the rabbi that he had done it, the rabbi said, “Now, go and gather the feathers. Because you can no more make amends for the damage your words have done than you can recollect the feathers.” 
In this version, the rabbi himself is the object of the gossip/loshon ha-ra and he gives the same feather-directions to the man (not woman) who comes asking forgiveness. I especially liked that the version on this site explicitly tied in the “ma’aseh” (story) with the Jewish concept of “loshon ha-ra,” even citing traditional Jewish laws about it. It’s good pedagogy!
In fact, when I googled the words “rabbi feathers,” I found innumerable places where the same story was told about a rabbi! 
Interestingly, I also found the following variation of the same story in poem-form:
A woman whose tongue was sharp and unkind was accused of starting a rumor.
She was brought before the village rabbi protesting, “What I said was in jest … just humor!
My words were carried forth by others. I am not to blame.”
But the victim cried for justice, saying, “You’ve soiled my own good name!”
“I can make amends,” said the woman accused,
“I’ll just take back my words and assume I’m excused.”
The rabbi listened to what she said,
and sadly thought as he shook his head,
“This woman does not comprehend her crime,
She shall do it again and again in time.”
And so he said to the woman accused,
“Your careless words cannot be excused
until …You bring my feather pillow to the market square,
cut it and let the feathers fly through the air.
When this task is done,
bring me back the feathers…every one.”
The woman reluctantly agreed.
She thought, “The wise old rabbi’s gone mad indeed!”
But to humor him, she took his pillow to the village square.
She cut it and feathers filled the air.
She tried to catch. She tried to snatch.
She tried to collect each one.
But weary with effort she clearly discovered,
the task could not be done.
She returned with very few feathers in hand.
“I couldn’t get them back, they’ve scattered over the land!
I suppose,” she sighed as she lowered her head,
“Like the words I can’t take back, from the rumor I spread.” 
So, is it a “Christian” story that is irrelevant to Judaism? Apparently not. Rather, it’s a story that seems to have had a Christian source that, as often happens with “folklore,” became “borrowed” or adapted and “claimed” by others for homiletic/educational purposes. Could it have started as a “Jewish” or “Hasidic” story and then become absorbed into the legends about St. Philip Neri? I don’t think so. I’m not enough of a folklorist to determine that, but I’m inclined to accept it as received: A Christian story that has been adapted to Jewish teaching.
The main thing to notice is how well the same teaching fits into both traditions.
It’s also particularly relevant in this era of online gossip: One statement, sent across the whole world in a matter of seconds, can never to be withdrawn.
In some other versions of this story, it is a novice monk rather than a woman who comes to the saintly teacher. There are also versions where St. Philip Neri tells her to scatter from a high balcony the feathers found in a pillow, then try to recover them.
I made some minor changes to simplify the story as I found it here (e.g. substituting “gossip” for “calumny” in St. Philip’s final statement.)
 Bard, Mitchell G.; “Issues in Jewish Ethics: Speech and Lashon HaRah”; 2016
citing: Forest, Heather; Wisdom Tales from Around the World
(August House, © 2005); pages 67-69
(I edited the layout of the verses to clarify the rhymes).