You need to pass the mission "From Zero to Hero" to open up the other island. After that mission you can unlock Drugs, Smuggling and Robbery businesses by completing an attack on a rival site of that type.
You don't get to unlock drug business until Armando Mendez tells you to attack one particular business site owned by bikers, next to the North Point Mall.
Edit: from my playthroughs with the game, once you unlocked the second island and attack or buy any new site, you'll automatically be proposed both Robbery and Smuggling operations, but Drugs will remain locked.
I remember this because I always thought it was weird that the Drugs empire isn't available even though the others that are more expensive and profitable already are! Therefor having a drug business just became useful only for its side mission and vehicle.
In any case, the issue here seems to be that the TC did not located the next Story Mission starting point, and has been busy taking over all empire sites of the west island instead.
Edited by Grievous, 30 September 2013 - 07:40 PM.