I have a question.
Can all the professors in a department see a particular student's grades?
Can the chair of a department see a student's grades?
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Officially and in general, no.
The teacher of a class should in general not share grades with his colleagues. Informally, however, there might be some communication of individual students' performance in ways that don't violate confidentiality: "she did very well on my class"; "he struggled on the exams, but did a very good job on the final project." Typically, this would be in the context of a colleague inquiring about hiring a student as a student worker or as a graduate student. Aggregate information could also be shared without harm to enable better matching of teaching to student needs and abilities.
That said, there are exceptions to this rule. For instance, your academic or research advisor will typically have access to all of your grades. Similarly, an evaluation panel—such as one that convenes for a graduate student's qualifying exams—will probably have access to all of the student's academic record.
But a random professor generally will not see a student's grades in all courses. This is by design and a logical move.
This seems to have changed somewhat recently at our school. I used to only have access to the grades of my advisees and perhaps current students. For the last year or two though, I seem to have much wider access.
A student from another department stopped by to talk with me about possibly attending grad school in our program. I looked him up in our computer system, and could see his whole transcript (which classes were taken in which semesters and the course grade for each). I've never looked up transcripts for students who I don't somehow know (either my advisee, or in a class I'm teaching, or at least someone who has asked me advising questions), but for the last year or more I've had access to every student transcript that I've looked for.
At the college I taught at, it was forbidden for anyone to reveal grades to anyone. The only people who had access to grades were the teachers who taught the course in question (I could see your scores only for classes I taught you) and counselors. Other teachers did not have access. Of course, some teachers did share grades informally.
In the Vietnamese university I teach at now, grades are posted publicly and actually emailed to every other student in the same cohort and teachers freely share grades with each other because they are not considered private.
In short, it depends on the rules and regulations where you are in addition to the informal relationships around that institution.
As others have mentioned this depends upon the place where you are.
In Spain, at the end of each term the grades of all the students registered for each of the available courses are publicly posted on a board.
I suspect it might be similar elsewhere in Europe. I remember seeing similar grade spreadsheets posted on a board here in Finland.
Another variant of the various models described in these answers, to illustrate the vast diversity in procedures and how there is no globally correct single answer to the question:
I am most used to exam corrections being community efforts. That is, when a (written) exam of a class by one professor has taken place, the whole department of that professor will be asked to help checking the exams. This is so the (substantial!) workload of correcting 50+ exams is shared.
In effect, it usually means that a considerable part of the PhD candidates of the department will do the checking, and often (as one person checking one particular task across all students is more efficient than one person checking all tasks only for a small set of students) this means that each of these PhD candidates will have seen the overall performance of each student in the exam.
Summary: In some places, it can be assumed that everyone within one department knows about student grades in some way obtained from that same department.
As a rough guideline, this can easily mean some 20 people or more were involved.