The Encyclopaedia of Islam has some good information on this. The basic answer is that it's extremely complicated and varied to place to place. I've quoted the relevant bits:

"There is frequent mention in histories of the repair of old ḳanāts, and the making of new ones by rulers and others. The upkeep of irrigation works (which included ḳanāts) was, indeed, one of the traditional functions of the ruler."

"Sometimes the ownership of the ḳanāt is in the same hands as the land which it waters. Sometimes the different parcels of land have a prescriptive right to the water of a ḳanāt with or without the payment of dues. Frequently the ownership of the land and of the ḳanāt is in different hands and the water is bought and sold. Many ḳanāts are waḳf property or k̲h̲āliṣa (cf. a nis̲h̲ān granting tax immunities to Fayḍābād, Balk̲h̲, which states that four dāngs of any ḳanāt brought into operation belonged to the dīwān"

"The distribution of the water of a ḳanāt is based on time. The rotation period varies from ḳanāt to ḳanāt, and the order in which the different parcels of land receive the water is sometimes permanent and sometimes settled from year to year by the drawing of lots or some other method. Normally land higher up has the right to water before land lower down. The shares are often fragmented into very small units of time. If the flow of a ḳanāt is considerable and those having a right to the use of its water numerous, the distribution of the water is likely to be under an official known as the mīrāb , who is chosen by the joint users, or, in the case of a k̲h̲āliṣa ḳanāt , by the government, and is paid by a special due."

"The maintenance of a ḳanāt in the case of joint ownership clearly poses certain problems, but there are numerous instances of ḳanāts owned by over a hundred persons being successfully operated. A substantial body of law concerning ḳanāts, based partly on custom and partly on the S̲h̲arīʿa , has grown up over the centuries."