As a marker, there are always some students who will email requesting to meet as they disagree with the marking. According to the professor, the marker should solve students' marking issues but I am uncomfortable with those students who insist that marks should be given in their way. Some of them are disrespectful. If I refuse to meet them, they will complain to the professor about me not engaging with students and the professor will criticize me. Should I engage with these students?
According to the professor, the marker should solve students' marking issues
There you have it. Yes, meeting with students and explaining your reasoning is one of your job duties. Even good graders make mistakes, and even good students sometimes don't understand where they went wrong.
I am uncomfortable with those students who insist that marks should be given in their way.
You do not need to negotiate with students. You need to explain why they were incorrect, and explain how the point deduction was calculated. (Hopefully you've been using a rubric so that students are consistently docked the same number of points for each offense -- if not, you should start.)
When students try to "insist," you can explain that it wouldn't be fair to adjust penalties on a case-by-case basis. You can be pretty blunt here: "I provide these meetings so I can answer your questions. I've already explained why you lost points; I am not going to debate with you on every question."
Some of them are disrespectful.
You certainly don't need to tolerate this. "I find that statement very disrespectful. If you continue to be disrespectful, I will ask you to leave." If you do in fact ask a student to leave, or not to return, you should pre-emptively send the professor a note concisely explaining what happened (e.g., "FYI, I told John he was not welcome in my office after he called me a _.").
Almost everyone is uncomfortable with confrontation. Being an adult means being uncomfortable when you have to.
My announced policy was that if I had made a mistake, I'm happy to change it. If not, we're not going to have a discussion about harshness. They love the word "harsh," and no matter how little you deduct for an error, you're still "too harsh" for their taste.
Nope, the sentence I used was, "Everyone gets screwed the same. That's fair." You get better grades by being a better student, not because you're a tenacious negotiator. In extreme cases, I would eventually say, "OK, at this point you're cheating and I'm about to turn you in to the Student Judiciary." What? "It's academically dishonest to try to get an unfair advantage over your fellow students. So it's cheating." Game over.