On Ilum there are a total of 5 Datacrons and all of them can be acquired by both Republic and Empire players.
Datacron 1 - Republic and Empire
Red Matrix Shard
Codex Entry 85: The Purge of the Dark Council
Coordinates: 543, 544
For this Datacron you need to go to the Republic base located in the Eastern Ice Shelf. The Datacron is at the top in a small tent. There is a Republic Elite that patrols near the tent so you may need to wait until he gains some distance before entering it if you're an Empire player.
Datacron 2 - Republic and Empire
Codex Entry- Galactic History 83: The Republic Rebuilds
Coordinates: 924, 1076
To get this datacron you'll have to go to the south-eastern corner of the Eastern Ice Shelf at approximately 941, -10 and move forward entering the exhaustion zone(will take about half of your life). To reach this datacron you'll also have to run through a snow storm and Republic players need to be careful as a heroic area with Imperial Elites is located north of the Datacron.
Datacron 3 - Republic and Empire
Codex Entry- Galactic History 84: The G0-T0 Coup
Coordinates: 103, -78
Go to the center of Ilum to approximately 265, -125 and you'll see a canyon where the Datacron is located. To descend into the canyon can either jump from ledge to ledge or take an elevator (the eastern one for Republic and western one for the Empire).
Datacron 4 - Republic and Empire
Yellow Matrix Shard
Codex Entry: Galactic History 86: The Empire's Revenge
Coordinates: 311, -377
The Yellow Matrix Shard Datacron is located in a building outside the Imperial Base and is guarded by Imperial troops. However, Republic players should be able to get to this Datacron easy.
Datacron 5 - Republic and Empire
Codex Entry- Galactic History 87: A Wayward Apprentice
Coordinates: 304, -1569
This Datacron is in the northeastern area of the Eastern Ice Shelf. To get there you'll have to go down on a slope and into an icy chasm that you'll have to cross. At the other end of the chasm there is a cave. Go inside the cave all the way to the northern end where you'll find the Datacron behind a large green crystal.