Can I stop people from being able to publish Notices in certain areas?

The nature in which notices work means that they are not area specific. Notices are content specific and someone who has the security role to manage notices can manage them on all content they have permission to see. It’s important to remember that security roles for content in an area will override the ability to manage notices in that area. For example, it would be impossible for a user to create a notice on a piece of content they don't have permissions to see.