Doom does things very differently to how more recent multiplayer FPS multiplayer works. Every player's host machine has identical state. That's not as superficial as things like score, who is alive/dead, etc - it's exact positions or all entities, their metadata... all the way down to the next number to be picked by the random number generator.
This is updated every tick (30Hz? can't remember) and every machine needs to synchronize with every other within that tick, or the game slows down. If you've ever played Doom over a modem, you know what it's like when someone has bad settings which result in high latency - in this case, poor frame rate. The data sent between machines consists of the inputs - e.g rotation, movement, fire, talk - and not higher level constructions such as motion vectors.
So it's entirely necessary that the "random" number generator is not so random at all. It's fairly hilarious to see the results of messing with it, though :)
It does make me think that cheats (not that I ever cheated) were missing a trick. If they knew what the next number to be pulled from the random number generator was, they could, for example, delay firing a weapon by a frame or two, to get more damage. Probably other more interesting things I can't think of immediately. If there was ever a bot vs bot Doom competition, this is something you'd definitely want to consider.