I have been a leased employee (temp through temp agency) at a client company in Connecticut for over 15 months. I definitely meet the IRS's definition of "common law employee", and also Connecticut's ABC test.
There are some circumstances where there may exist what is called "dual employment." That said, most companies use contractors to fill out the manpower needs so that they don't have to worry about the paperwork involved with employees. Your contract of employment is with your temp agency and that is who will provide whatever benefits you receive.
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If you are talking about health benefits or workers' compensation benefits, the agreement between the leasing company and the purchasing company will decide which company is responsible to maintain benefits/insurance for you, if you meet certain requirements.
In my experience, a manpower agency typically carries workers' compensation insurance to cover temporary/leased employees, and an injured temporary worker is not barred from bringing a lawsuit against a temporary employer for injuries sustained on the job.
If you have more specific questions, feel free to give me a call.