Can I share public images of someone who is breaking the rules?

No, that is not allowed. As an entrepreneur or citizen, you may be confronted with theft or nuisance, for example. And you may have (security) footage of that person. It may seem useful to share that footage in order to track down the offender or nuisance maker. For example, on social media, in a chat group, or at the entrance to a store. However, public shaming can have a major impact on someone's privacy, especially if that person turns out to be innocent. That is why it is prohibited under the GDPR. If someone breaks the rules, it is up to the court to determine whether they are actually guilty and to impose a penalty. Are you an entrepreneur? Then there are other things you can do to warn your staff and other shopkeepers about a (convicted) offender. For example, you can keep a blacklist. But you can only do this if you meet the conditions. See: As a shopkeeper, am I allowed to display or share photos of shoplifters? As a citizen, do you want to tackle an offense or nuisance? Then go to the police. Do not share images on your own initiative. Incidentally, the police may only share images publicly in exceptional circumstances.