I had similar thoughts reading the rules & came to the same conclusion. Doesn't look like there's a way to trade/give weapons to other marines either. So I guess you just avoid eg. one marine collecting them all, since they wouldn't be able to make best use of them (limited actions + deck dilution).
This was also a significant factor of classic DOOM co-op. Only one person can pick up each weapon, so if you hog them all, your buddies are going to have a hard time keeping up with the tougher demons.