The shifts you work as a uPaged nurse are casual shifts. Every casual nurse who takes a shift at a hospital must sign a casual contract with that hospital at the first time of their first shift, so that you can get paid via their payroll and covered by their insurances. Even when you are there to work for only one shift and may never work there again, it must be signed. You won't need to do this for subsequent shifts.
As a casual employee of that hospital via uPaged, you will be booked first and cancelled last by that hospital, and enjoy other benefits of being a casual employee of that hospital (ie: training and education, salary packaging, professional development etc).
Even though the hospital becomes your employer, you choose the shifts with the rate of pay you want, and you will always be a uPaged nurse as long as you continue to book shifts through the platform.