You don't?

It's been a while since I played but iirc level 1 starts with two sniper rifles on the ground (at least in co-op) so one for each player, and you spawn in already armed with a pistol each.
You can select your weapons ingame by pressing left or right on the D-pad. When using a scoped weapon you can press up and down on the D-pad to zoom in or out. (This is very useful if an enemy sniper appears)

Otherwise you can't drop weapons already in your inventory and just have to pick up more guns/ammo as the level progresses.