This is a parallel circuit.
Therefore, when the first person presses the button, it will close the circuit which will light up the bulb.
The current flow from the longer (positive +) side of the battery. And the amount of current flowing through each of battery is the same.
When one bulb lights up after one another, the brightness of each bulb will decrease.
When all the switches are closed, the brightness of each bulb will remain the same.