In cases like this, it is helpful to look at a schematic of what you are trying to do.
Notice that pin2, pin3, and pin4 are all effectively on the same piece of wire? So when one of the buttons is pushed, they will ALL see 5V. No, you can't use a "shared" pull-down resistor.
However, why not use the internal pull-up resistors instead? Then you don't need any external components. The only trade off is that the logic is inverted (unpushed buttons are HIGH and pushed buttons are LOW.)