Zakat is to be Collected and Distributed by the Khalifah

Narrated Abu Huraira: When Allah's Apostle died and Abu Bakr was elected as a Caliph after him, some of the Arabs reverted to disbelief. Umar said to Abu Bakr, 

"How dare you fight the people when Allah's Apostle said: I have been ordered to fight the people till they say: None has the right to be worshipped but Allah, and whoever says: None has the right to be worshipped but Allah, saves his wealth and his life from me unless he deserves a legal punishment justy, and his account will be with Allah!" 

Abu Bakr said, "By Allah, I will fight him who discriminates between Zakat and prayers, for Zakat is the compulsory right to be taken from the wealth by Allah, if they refuse to give me even a tying rope which they use to give to Allah's Apostle, I would fight them for withholding it." Umar said, "By Allah, it was nothing, except I saw that Allah had opened the chest of Abu Bakr to the fight, and I came to know for certain that was the truth." (Bukhari: vol 9, bk. 92, no. 388, Khan)