As you play co-op you will earn points for getting kills, assists, vehicle destroys, etc. When you hit so many points you unlock a weapon. If you complete a mission successfully expect 13-16k points depending on how good you were.
MP412 REX - 63,000 score - All kit revolver side arm.
KH2002 - 126,000 score - Assault kit assault rifle.
MP7 - 189,000 score - All kit SMG.
M39 EMR - 252,000 score - Recon kit rifle.
93R - 315,000 score - All kit side arm.
SG553 - 378,000 score - Engineer kit carbine.
G3A3 - 441,000 score - Assault kit assault rifle.