What is the work authorization for P-1S visa holders? 

Is it possible for P-1S visa holders to directly work for a training facility?

Generally no, P-1S visa holders cannot perform work directly for petitioners on tasks not associated with the principal visa holder. However, if the principal P-1 visa holder is doing ancillary tasks for sponsors and stakeholders, the P-1S visa beneficiaries may assist these tasks. 

If a P-1S visa holder wishes to work directly with a training facility on tasks not related to a principal visa holder, then this applicant should try to apply for their own O-1A visa application. 

Can a P-1S visa holder work at multiple training facilities not associated with petitioners? 

No, the P-1S visa is not appropriate for this purpose. The current visa type is an O-1A visa with the appropriate agent based structure to support these activities.