It's advisable to share the authentic information. I think we need to be responsible what we share.
In case of harming the reputation I think it depends on what the person shared. There is nothing wrong also sharing the real fact even if it harm the reputation of the company or school. The real fact is who's fault cause the trouble.
I agree with you that it is okay to share the fact but I disagree with your statement of even if it will harm the company or school, are you saying if you have a company and one of your employee is spitting fact that could make your company go down, it is okay and you will let him be?
If it's my company I won't want to see such a thing and I will dismiss the employee as I said world is not fair. But if I done something wrong and a employee shared it, it's my fault and I should not punish him if I think about justice.