Cohorts are NPC's. (Page 104 of the DMG)
The DM controls NPC's. (I have no idea where this is stated, but I guarantee its truth)
As far as Leadership goes, the Cohort is loyal to the PC beyond just money or convenience. The Cohort believes in the PC on a deep enough level that if the PC tells the Cohort to attack a goblin, it will attack that goblin, trusting that the PC won't lead him astray. This level of obedience is similar to Domination, and extremely inconvenient and inefficient for anyone but the PC themselves to roll-play, and so for practicality, the PC simply plays the Cohort as well as his own character. What is the point of having the Player tell the DM what to make the Cohort do each round, and add rolls that the DM has to make without it changing anything except taking longer?
It of course IS up to the DM though. It is not stated anywhere that the Cohort from Leadership must be played by the PC, but it is stated that the PC has a level of control over the Cohort as to be able to control mechanical aspects of the character (skills, combat, etc.).
I personally like to play the Cohorts of my players, as it gives the Cohort a personality that most players usually wouldn't give to it, while still letting the player decide when the Cohort makes skill checks, cast spells, etc.