Not only should it be required that all accidents or incidents be reported, they should also be evaluated by the departmental safety committee and discussed at departmental safety meetings.
While having no accidents isn't necessarily a good indication that everything's okay aving them go unreported makes the matter worse. The reporting of all accidents is extremely important. Every accident is an opportunity to improve your safety program, to learn how to do a better job, and to protect your workers and facilities. Since accidents happen relatively infrequently in smaller organizations, keep track of the incidents and close calls as well. These are the events where matters of inches or seconds were the difference between nothing happening, a minor mishap, and a major disaster. The rule of thumb is that there's three hundred minor incidents for each major one. Think of all you could learn from having a chance to review the close calls.
The safety committee should get copies of each accident or incident report and review it carefully. They should conduct an investigation of the event so that it can be correctly evaluated and the proper corrective action taken to prevent a reoccurrence. Don't go around looking for someone to blame. Looking to place blame is the quickest way to convince people that they shouldn't talk about what happened, to avoid telling the truth, or to have a loss of memory.
Then, the event should be brought to the attention of the rest of the people in your organization at a departmental safety meeting or by other means so that they too can learn from the experience. Photographs of injuries and property damage are graphic reminders of the consequences of carelessness, unsafe work conditions, and unsafe work practices.
Consider having an accident/incident report form for your employees and students to fill out. It will help them to develop an appreciation for this recordkeeping aspect of safety.