I have decided to offer a second answer from a different point of view. That of the bully.
Years ago I went on a residential course in the US (I'm British). It was about personal development and I paid for it myself. It lasted for 8 weeks and the participants did a lot of sharing in front of the group.
One session was about "the worst thing you have ever done". A young woman stood up and said that she had been a bully at school. She picked on one particular girl whenever she could. She said that she had no understanding of the damage it was causing and did it because it made her friends laugh and made her feel clever and popular.
She had only come to realise the profound damage this caused to another young person after hearing another member of our course confessing their own experience of being bullied.
Some bullies, especially verbal bullies, are blind to the effect on the recipient, they simply revel in the acclaim and laughter of their friends.
Is it a good idea to admit your vulnerability to the bully? In my opinion this is a strong No. Outside of the long-term close-knit group situation, where others are listening, this could be seen as an encouragement to escalate. Be strong, not vulnerable.
- Me as the bully. When I was in my early teens I was annoying in class to teachers who could not keep good order. It was all verbal - silly noises, chanting, etc.
One day, when I was about fourteen, I noticed that our French teacher almost ran to the back of the room and stood behind us with his back against the wall. I turned to look and realised that he had tears in his eyes. This was the first time I had thought of a teacher as a human being with feelings. From that day onwards I observed him and came to understand that he was really enthusiastic about his subject and all he wanted to do was treat us like adults who wanted to learn. He would have done well at an evening class but he had no idea how to control unruly teenagers.
At that point I also started to notice that other teachers somehow had a presence that precluded anyone trying to make fun of them. I've never fully understood it but it was somehow a manner that simply expected good manners and would be disappointed with anything less. We were more interested in getting their approval than that of our classmates. I think it was because they showed their approval for good thinking in class.
- My father
He was not a bully, or at least not much of one. However he completely lacked any understanding of another's point of view. When I wanted to go to music college, he was testing me by playing simple tunes on the piano that I had to write down in music notation without looking. He grew very impatient with me at one point. In frustration, I said, "Let's swap". He was amazed to discover that he had as much trouble as I did. Some people are like that - they simply can't stand in the shoes of another.
- A woman at work
I came to think of her as evil. She would deliberately make fun of anyone who she perceived to be vulnerable - especially if they were going through a difficult time. Example: Someone said their mother had died and they had scattered the ashes in her favourite place. Her remark was, "Oh, you can never be sure you get the right ashes, they could be anybody's." She was acutely sensitive to weakness but had zero empathy.
I later discovered that she recorded conversations on her phone so that if anyone said anything nasty to her, she could complain they were the bully. I know of at least two people who left their jobs because of her.
Moral: Never try to use a true bully's tactics back to them. They have a lifetime of experience and you will lose, just as surely as if you are an amateur at chess playing a grand master.. Retain your own morals and standards.
The true sociopaths are rare. They will never give you credit for anything, in private or in public. Their purpose is to pull you down. There are two ways forward,
(a) Leave and avoid them as much as possible. If you stay nearby, they will still continue to undermine when possible
(b) Become sure of your own worth. This may not be at all easy. Keep your behaviour impeccable including being polite to the bully - but firm. Have a few phrases, e.g. "Please don't speak to me like that", "That's not very nice" or a fake laugh pronounced as words, "Ha ha" if they try to be funny.
Notice their criticisms and internalise them as feedback. Compare with your honest beliefs about yourself. If you know deep down that what you are doing is okay, then think that to yourself when necessary
They may simply be unaware of others feelings - help them to understand but without showing weakness. They may be a genuinely nasty person. Do not stoop to their level -- they will win. Behave impeccably but make a non-emotional response when they cross the line. Prepare the phrases ahead of time. You only need a few and they don't need to be funny - just an expression of disapproval as from a parent who expects better.
Treat the whole thing as a fascinating psychological experiment that does not affect you or your emotions. Observe their behaviour and your own behaviour as would a third party. Have fun analysing what is happening.
You may like to experiment with complimenting the bully in public when they do something good. This will give a pleasant shock to their system. If you combine this with mild criticism when they are not good, they will start to crave your praise. It's like dog training.
Above all, keep clean yourself. The more you define your standards and stick to them, the more you can expect from others. Don't lower yourself to anyone else's level.