The term "shared governance" seems to date back to a 1966 statement by the AAUP. It was apparently a push for reform and liberalization in the management of colleges and universities, which had traditionally been run in a more top-down fashion. The legal structure of a school is usually that there is a governing board, and the trustees on the board make general policy decisions but delegate day-to-day authority to college presidents. However, there are some areas in which faculty and students should not just be treated as serfs by administrators.
Faculty are the experts in their academic subjects, and therefore they are supposed to have primary authority over curriculum and grading. This authority is not absolute. At my school, for example, the physics faculty initiated a process to delete several courses that we thought were not needed, had not been offered in a long time, and had not worked out well in practice. To get this to happen, we had to shepherd the deletion through many bureaucratic steps, and we encountered opposition from the vice president of instruction, which we had to deal with through a long process of negotiation.
The AAUP statement also says that faculty should have "primary" responsibility for decisions about "faculty status," i.e., hiring, firing, and tenure. But "primary" doesn't mean that they control it completely. The responsibility is shared with administrators. For example, when we hire tenure-track faculty at my school, a faculty committee picks three candidates, who are then interviewed by the president, and the final decision is made by the president.
There are some areas, such as budgeting, where control would traditionally have been exercised completely by management but the AAUP statement recommends that faculty have a role because of their expertise in their academic areas.
Shared governance is generally expected to cover academic and professional matters, but not workplace issues such as paychecks and working conditions. If there is a faculty union, that would be the union's bailiwick. In practice, however, faculty senates may work fist-in-glove with the union on these issues.
Shared governance is not supposed to be a process where management accepts "input" from stakeholders and then makes decisions based on that input. That would be more of a corporate model.
For students, the AAUP's statement makes some very limited recommendations, which are clearly described in the document itself as a political compromise reached during a period of upheaval in American education (the Free Speech Movement and the Vietnam War):
The respect of students for their college or university can be
enhanced if they are given at least these opportunities: (1) to be
listened to in the classroom without fear of institutional reprisal
for the substance of their views, (2) freedom to discuss questions of
institutional policy and operation, (3) the right to academic due
process when charged with serious violations of institutional
regulations, and (4) the same right to hear speakers of their own
choice as is enjoyed by other components of the institution.
Note that it does not, for example, advocate giving them a right to speak freely on social issues. In 1965, one of Ronald Reagan's promises in his campaign for governor of California was to "clean up the mess at Berkeley." Since then, things haven't changed very much. For example, my school does not allow people to gather signatures on a petition except by standing on one specific patch of grass, and they have to register in advance and state what the petition is about.
For public schools, some of these issues may also be regulated by state laws. For example, in California, title 5 of the state ed code requires governing boards to "consult collegially" with faculty senates and to make room for "effective participation" by faculty and students. California state law also severely limits the circumstances under which a grade assigned by a professor can be changed against the professor's wishes.
How does it work, on a practical level? (I understand what a session of the U.S. Senate looks like, I have no idea what a session of a faculty senate looks like.)
Elected faculty representatives from various departments meet and discuss items on an agenda. For the most part they can only make recommendations on policy -- they can't set policy directly. They can form committees and decide to accept or not accept the committees' reports and decisions. There are also advisory bodies, e.g., at my school we have a President's Advisory Committee, and one of its members is the faculty senate president.
What are the benefits and disadvantages of this approach to governance?
Compared to a more top-down decision making process, it's intended to protect against political or short-sighted interference in academic matters. For example, it's supposed to prevent administrators from firing a professor who has controversial social views. Anecdotally, the main disadvantages I've seen are (1) that the quality of the decision making in a faculty senate is often extremely low, and (2) that the faculty senate may not show sufficient independence from the union.
What kind of hierarchy typically exists? (Who reports to who?)
It's a structure that bypasses the hierarchy defined in an org chart. Faculty senators don't report to anyone. The org chart only includes management.