When you work in a high-pressure or high-risk environment, safety has to be a number-one priority. Whether you're an entry-level employee or the owner of a company, you're going to have something to contribute to workplace safety. That said, the duties related to safety will vary somewhat depending on the title you hold in the company.
The leaders of an organization are the people who make company policies -- and that includes the safety policies. If you're in an industry that mandates certain safety equipment or practices, it's the employer's responsibility to make sure that those guidelines are followed, and that employees have access to the information they need to implement those safety mechanisms. This includes training and government-mandated safety posters, but it may also include a safety brochure or information included in the employee handbook. Additionally, employers need to keep records of injuries, and report serious health issues to the Occupational Health & Safety Administration, or OSHA.
In many workplaces, the higher-ups set the policies, and the managers ensure that they're followed. Managers or supervisors are often the ones to witness the day-to-day operations of employees. When a piece of equipment breaks or an employee violates a safety policy, it's typically the manager who deals with it -- either reporting the problem to his superior or fixing the problem. This goes back to the company policies; some companies have a "top-down" model that requires all problems to be dealt with by the company leaders; other company policies allow managers to run their departments and dole out solutions and punishments as they see fit.
When a company has safety policies in place, there needs to be a mechanism by which employees learn them and get updates. In some companies, these duties fall onto a safety trainer or inspector. Factories with large equipment or complex construction projects may have a dedicated "safety person" to whom the responsibilities of monitoring and training fall. Trainers or inspectors may work closely with employees and let them know when they're violating policy, or they may be managers-trainers who wear different hats as needed.
The workers in any company also have a responsibility to follow safety procedures and policy. This includes wearing the safety gear provided and following the company's guidelines -- but it also means reporting fellow employees when they violate safety rules. Workers will also be responsible for attending required training to keep a company in compliance with mandated safety regulations.