For clarification: schools run a background check from the DPS database. If that comes back "clear", the school check references. Former employers are only technically allowed to provide 1. dates of employment 2. eligible for rehire yes/no 3. they may also provide safety info for DOT sensitive positions.
Anything beyond the above listed information can expose the former employer to legal trouble. If they reveal unsubstantiated or unverified rumor type information that prevents a former employee from being hired, they can be subject to lawsuits.