As is clear from many of the comments on my last post, there are many different structures for how grad students get matched with advisors. There are pluses and minuses of each system, but I've been happy with the system we have; i.e., grads work with a specific advisor from the start. That means that from the start, I am responsible for helping design the best course plan and for getting the student started on a research project, and I like that.
Of course, the students are free to change advisors once they get here, as long as there is another advisor willing to take them on and they are in good academic standing (based on evaluation by the graduate committee). Advisors are also free to terminate their status as advisor with a particular student, although these decisions are typically made in consultation with a faculty committee.
I have co-advised students in other departments, including engineering departments, where the grad-advisor culture is very different. In one department where I have co-advised, student aren't matched with any particular advisor when they first arrive, but they choose in the spring of their first year. After that, all students are supported in full on grants, but they spend part of the time as a TA while they are being supported by their advisor's grant. I wasn't too happy with that, as my students spend a lot of time teaching while supported by an RA on my grants. Faculty in that department typically have lots of industry grants in addition to NSF grants, so it is probably easier in general for them to manage continuous funding whether or not students are TA's are RA's.
I prefer the system we have, in which my students spend some time as TA's (supported by the department), some time as RA's (supported by my grants and not doing any teaching), and (for the most outstanding ones) some time supported by fellowships. Another option is for a student to teach half the usual load and be half supported on a grant.
It's not a perfect system. Unlike the system used by the engineering departments here, there is more inequity. Some students have advisors with little or no grant support -- these students spend most or all of their time as TA's and may not be supported in the summer (though the department has some funds to give them some relief for a term or summer, and there are summer teaching positions). Other students are very well supported, including in the summers.
Even so, I like the present system because it has a lot of flexibility for both advisors and students. Students get a range of experiences, and we can adjust the balance depending on where they are in their grad program with respect to course load, exams, travel, and other considerations. I can also figure out in advance how many new students I want to accept to my group based on how many students are graduating, how much grant support I have, and so on. That's what I'm in the process of figuring out now for next year.
6 years ago