Our company uses a lot of "1099 employees." I was recently told that's not permissible. What's the deal?

By definition, someone receiving a 1099 is not an employee and is an independent contractor. If a company is referring to workers as "1099 employees" something is wrong with the classification system for the company.  The company has an affirmative duty to ensure that the workers either meet the relevant legal test to be independent contractors or they need to be converted from 1099 workers to company employees who receive a W-2. Note that misclassified independent contractors are an enforcement priority for the U.S. Department of Labor.