Every city has a public art commissioner, and in most cases they are the one who works with whatever organization the artwork is for to set guidelines (theme/location/size/funding/safety/etc) and commissions the artwork (hence the title). There are private commissions by developers for private properties (say, in front of a building or something), but the public art commissioner at the very least will know about it if not approve it. If its on a private property and does not in any way affect urban spaces (like hidden in the back yard or something), then you do whatever you want as long as it does not kill people or disrupt neighbors. 

There are a variety of ways in which an artist can be involved in public art, there is no one specific way. But yes if it is over a certain size it will definitely involve some kind of contractor / engineer / fabricator / inspector relationship with the artist, sometimes appointed by the city, sometimes through the artist. It is hard to provide you with a concrete answer because it really depends on the circumstance / context. If you have a specific example then perhaps it may make your search for answers more fruitful. For Turrell, some of his work is essentially architecture, so it probably came into existence the same way a piece of architecture is built to reflect the architect's vision.

In any case, if your artwork reaches the point where it needs to be fabricated by someone else, then you are essentially a "creative director" of some sort. Again, vague title unless more context is provided. Hope this helps!

Specialty fabricators or by the artist. Permits, insurance, etc. are a different matter.

Hsiuchi, thanks.  I guess there is a fine line between creative director and general contractor correct?   

The difference is really in the legal definitions. It has to do with payment, responsibilities and liabilities outlined in a contract. And again, every situation is different. Whatever your contract needs you to be thats what you are.