The key factor as to which entity must submit the 300A data is the same factor that has always dictated which employer, the host or the staffing agency, is responsible to record the workers’ injuries. That factor is, which employer is supervising the day-to-day work activities of the workers. If the host employer is providing direct day-to-day supervision; i.e., giving assignments, overseeing the work, providing the tools and safety equipment, etc., then the host is responsible to record any injuries to those workers. If the staffing agency provides its own on-site direct supervision, then the staffing agency is responsible to record the injuries and submit the data.
It is not really an issue that can directly be contracted away. The contract could set the terms for that level of supervision, but who records and submits is driven by the reality in the field.
Here’s some guidance OSHA has issued regarding recordkeeping responsibilities for temporary employees.