Each player spawns his own enemies.
On the server, the spawn rate and the maximum spawns are determined for each player. These have varying values depending on whether it's day or nite, the layer you are in or being in any of the special zones like the Corruption, Dungeon, Meteor or Jungle; not to forget about the Blood Moon...
The harder the time, depth and zone; the harder your adventure.
They don't spawn each X seconds but rather by a chance on each single game tick up to a limit.
The spawn rates and maximum spawns from high to low for the same time and layer depth are
Dungeon, the skeletons must kill you! (not influenced by depth and time)
Blood Moon, it's an invasion... (not influenced by depth)
Jungle, to make it harder than other zones by sending those hornets.
Corruption and Meteor
World, which also covers the Underworld.
Thus, depending on when and where you are you can get different order in the lower three. The trend is that you get more spawns in the night and the middle of the map as you'll see zombies, skeletons and all kind of slimes annoying you on your path...
Please note that you can also influence this yourself, increasing chance by 33% and limit by 50%.
How can I test this out?
Is it really about chances?
This is a hard one given that it's works based on a chance and thus could result in a different outcome. A good example of the chance can be seen when chasing down a dungeon without killing the Skeleton first, you'll see a random amount of skeletons spawn after a random amount of time.
On my first attempt I ran down and took the contests of a chest, on another I got killed by 2 of them.
Do the time, depth and zone really affect the amount of spawns?
This is easy to verify, check the amount of slimes that visit you when you stand still during the day compared to the amount of zombies that hunt you during an invasion or even just the amount of fire balls that annoy you when you are mining a meteorite.
Do we really get more spawns together?
The best way would be to play together with a lot of people to drastically increase the amount of spawned monsters, playing with 8 players together on the same screen should give a noticeable difference against just 1 or 2 players. I've been mining a meteorite with my sister and noticed that we each got separate fire balls hunting both of us and when she died I only saw the half the amount...