I do not fully understand which interest a supervisor has in their student's work. Is the Ph.D. student somebody who the supervisor should teach, a colleague, an employee, or an independent researcher who is given advice when and how they feel like?
That depends on the field. Often (but not always) in the sciences the supervisor will have a number of research themes on the go, and a PhD student will slot in to one of those and act as a junior member of a team working on a larger problem, becoming more senior as they gain more experience. In the arts, and mathematics, it is more common for the PhD student to be closer to an independent researcher from the start. There are no hard and fast rules though.
The ideal is for the student/prof relationship to involve a degree of professional trust. They should each be striving to give the other any information that is required to proceed correctly. There is a lot covered by that, and many complex situations arise. But the usual principle should be that everybody should be striving to treat the other person with due respect and consideration. If they don't, it's a problem of some kind.
The exact nature of the relationship will depend on the university, and the specific department in that university. Different schools have different standards and cultures. The way a PhD candidate "fits in" to the school will be very different. Even individual profs will treat their PhD students very differently.
This is especially true when a particularly famous prof has many students so that they can say "trained under" about the famous person.
I recall a story about the Great Man of Science (GMS) and how he ran his group. One of his PhD students gave a talk to the department on the progress of his research. And GMS was in the front row of the audience, asking interesting and useful questions. And everybody was quite happy with the results of the talk. And at the end of the talk the following occurred.
GMS: That was a good talk. Your research is quite interesting. Who is your supervisor?
PhD student: You are, Dr. S.
GMS: Oh! Oh! Well, we should talk tomorrow, this is good work.
As well, to some extent, it will depend on the kind of relationship the PhD student desires and makes possible. If you want a more direct relationship, with more interaction, it may be possible you can arrange this simply by asking for it and showing up in the prof's office more often. If you need financial support you may be able to get this by asking for it. Maybe, if the prof or the department has funding. You might need to do TA work or something. If you prefer a less "hands on" type of relationship, you might be able to arrange that. It would depend on how good you are at working on your own with infrequent meetings and guidance.
It's good if the relationship is friendly and respectful but often the relationship between Supervisor and Ph.D. students is essentially a master / slave relationship.
I was lucky enough to have a supervisor who was kind and respectful so in my case it was more similar to the relationship between two employees in a company where a more senior employee shows the junior employee how to do things that he needs to do in order to complete his work but otherwise basically just leaves him alone and lets him work on his own.
Someone I know well had a supervisor who used the Ph.D. students as slaves to write papers for him. He refused to let her graduate, even after she had exceeded all the requirements, because she was producing papers that helped his career. He did that to all his students and they had to lobby the head of department to force the supervisor to let them graduate.
Professors at some universities are under tremendous pressure to publish. When they have Ph.D. students, who are also under pressure to publish, and those students are required to write the supervisor's name as a co-author on their papers, it's like they get papers for free without having to work for them. Of course some Ph.D. students require a lot of support from the supervisor so it's not necessarily "free" but in the case I cited above, the supervisor would just give a suggested research topic and then leave the students on their own to write the papers and do the research. If they spent a year getting no results, or if they failed out of the program because they didn't publish anything, it wasn't the supervisor's problem.
My ideal supervisor (from the student perspective) is someone who has great ideas about what research to investigate. He or she assigns you a topic or gives you a choice of several topics to investigate. He or she then suggests some journals articles to read to get up-to-speed on the topic. Once you've done the reading, the supervisor gives some suggestions about experiments you might try, gives advice about how to conduct the experiments, gives suggestions about the statistical methods to use to analyze the results, and then suggests a journal or conference in which to publish. After submitting, a great advisor would consult with you about the feedback you got from the journal reviewers and advises you about how to proceed (fix your paper to follow the suggestions, try publishing in a different journal, or abandon it and work on something else).
Basically, a good advisor is someone who is more helpful and less selfish. And it's important to get a good one because a really selfish one can make your life very painful. When you join a Ph.D. program, you're committing 5-8 years of your life to it, and if you fail, you probably won't get another chance. So if you're 4 years in and you realize that your supervisor is taking advantage of you in a harmful way, you have a serious problem. You can apply to change to a different supervisor within the same school if you have proof that the supervisor is doing something wrong, but if he or she is just being cruel and selfish but not actually breaking any rules or laws, you might be forced to decide whether to accept a kind of temporary slavery or drop out of school. That's a hard decision and most people I know chose slavery.
It completely depends on the advisor. I've seen treatment range from "colleague who I check in with once every month or so" all the way to " 'employee' who should do exactly what they are told without 'talking back' ".
Your program likely has certain standards or expectations (either implicit or explicit) around how PhD students are to be treated. They are often found in the student handbook or formal program requirements if explicit. It is another matter entirely, however, if your dean or department head will actually take any action when a supervisor violates these standards.
I think the above comments are all correct in terms of the sentiment, and especially the one warning prospective graduate students to choose their advisor wisely is very very good (speaking as a PhD advisor myself).
But PhD students are /students/ first and foremost. The degree to which they take on employee-like characteristics varies by field a lot. In the US at least, in STEM fields there tends to be more of a sense of students being "employees" because very often (most of the time) the advisor is literally winning grants to pay the salary of the student to do some work on a federal grant which will want reports on that work. This can tend to make things more 'contractual' than they would be ideally. A good advisor will try to resist thinking about it this way, as much as they can, to give the student room to explore, make mistakes, and grow. But it's not easy given the pressures faculty are under as well. In Europe and many non-US settings it is different, as most PhD programs are government-funded (in STEM at least, which is all I can speak to), and because bachelor's programs tend to be focused (not liberal arts), so PhD students abroad will be more likely to be independent and have a more 'traditional' mentor/mentee relationship compared to the US.
I have met professors who think students working on their research are
- Cheap labor. But students would work their ass off and graduate in the end.
- Assets: each graduate student who works with them will add more to the body of research, and in a certain way make said professors look good. So, both win.
- Cheap tools: a certain professor would get students to do not only the work but also as cheap secretaries. As the work was about to be published, and they were about to defend their thesis, professor would then have students start a new project. Then he would complete the last details, and write the paper without putting the student's names in them so he could also hold them hostage (you are not graduating until I decide you are ready to defend your thesis), specially given he only hired foreign grad students. Said professor has been known to call the immigration on the students if they disagreed with the arrangement.