When two or more firms share responsibility for the management or the "ownership" of an employee it is called co-employment. Typically, this can happen when a temporary worker (the employee of a staffing firm) works on a project for client firm. While it may be initially clear that the employee works for the staffing firm, during the assignment the employee may feel that they have received signals or instructions that they are actually the employee of the client firm.
The typical types of information that can lead to the employee believing this: having little or no contact with the staffing firm; placing the employee's name in the client's corporate directory; attending "employees only" events, meetings or parties at the client; reporting to a manager of the client firm; having their bonus and raise determined solely by the client firm; being on assignment for years at a time; etc. When a client firm acts as if an individual is their employee, in the eyes of the law that employee may become the responsibility of that firm.
Employers need to avoid being a co-employer, because a co-employer is a legally ambiguous position. In a co-employment situation, either "employer" could be responsible for the actions of the other. Either (or both) employers could be legallly/financially responsible for: payment of holidays and vacations, provision of benefits, continued employment, evaluation of work quality, promotions, etc. Even in situations where a contract clearly lays out the obligations of each employer, the actual condition of co-employment may supercede the contract.
In outsourcing, there is an exceptional need for a clear understanding, and a clear set of rules, that the outsourced workers are employed and managed by the outsourcer, not the client firm. The outsourcer must be the only "manager" of workers. This can be especially difficult if the program staff has been transferred from the client to the outsourcer. Even more so if the program is operated "on-site" in the client's space.