Organisations outside the government are allowed to use your BSN if this is stated in a law. And only for the purpose described in the law. In the context of hiring/forwarding, this is included in the Implementation Regulations Mandatory Use of BSN.
If the tax authorities make use of the hirer's liability, they determine the liability of the hirer/forwarding company based on a good administration. If there is no proper administration, the tax authorities apply the anonymous rate. This is the highest rate for payroll taxes.
Click here for more information about submitting (the data on) the identity document (Dutch only).