"I am the judge, and you are the lawyer."
Translation:Én a bíró vagyok, ön pedig az ügyvéd.
As it is a clarification (of the differences between the speaker and the listener) the personal pronouns are needed, this is an exceptional case, with a lot of implied meanings, not just a general sentence. You are also missing the definite articles. 'There is only the two of us of interest here and we each belong to two different categories'.
The sentence expresses a particular case where the subject pronoun are needed. Adding the subject pronoun is needed for emphasis and clarification. When you say "I am the judge, and you are the lawyer", it means "I am the judge (not you, nor anyone else)..." and this is translated to "Én a bíró vagyok..."
I'm not sure actually. The sentence expresses a particular case where the subject pronoun are needed. Adding the subject pronoun is needed for emphasis and clarification. When you say "I am the judge, and you are the lawyer", you really mean "I, and no one else, am the judge" and "you, and no one else, are the judge" (or "It's me who is the judge, and it's you who is the judge"). This "and no one else" is translated by the use of personal pronouns.
If you just said "A bíró vagyok és az ügyvéd vagy", it would sound very factual without any intend to compare you and the others. But then why use the definite articles a/az (as in "there is one judge, and it's me" and "there is one lawyer, and it's you")?
First of all, let's clear it up - "vagy" doesn't belong here. "vagy" is the singular second person indicative (present tense blablabla classification...) of the "to be" verb. It's "you are" for the "te" version.
For the rest, I'd say it's mostly a matter of keeping the distance. "Ön" is like distanced while "te" isn't. So with friends and or withing closer acquaintance, you are very likely to use "te" and singular second person forms. In an office or e.g with your teachers, you are likely to use the more distanced third person forms with "ön" (or "maga" or whatever, let's not get into half-obsolete nuanced formality details). With people who are considerably older than you, it's better to start off with keeping the distance, it's considered more respectful. In a shop or with colleagues, it's really a matter of many factors, the environment, the other participant, etc.