The Joysticks on the Atari 2600 can have the original stick and single button, a set of paddles, or even Mega Drive/Genesis controllers connected.

The buttons work by simply grounding the signal of the button/direction to ground to trigger pressed.

The fire buttons specifically (pin 6) get pulled up to 5V through pin 11 of the top board wire harness via resistors R225 and R226.

If you experience specifically fire buttons not working or intermittent, it is usually pin 11 on the ribbon cable between the top and bottom boards that is bad, but it is easy to test by measuring if your fire buttons (pin 6) has 5V when not pressed, and ground when pressed.