Ea are propriul birou.

You can often leave it out.

Romanian usually drops subject pronouns when the verb already makes the person clear. So:

• Ea are propriul birou. = She has her own office/desk.
• Are propriul birou. = Has her own office/desk / She has her own office/desk.

Including ea usually adds emphasis, contrast, or clarity:

• Ea are propriul birou, dar eu nu.
  She has her own office, but I don’t.

Are is the 3rd person singular present form of a avea, which means to have.

A few present-tense forms are:

• eu am = I have
• tu ai = you have
• el/ea are = he/she has
• noi avem = we have
• voi aveți = you (plural) have
• ei/ele au = they have

So in this sentence, Ea are... simply means She has...

Because propriu agrees with the noun it describes, not with the person who owns the thing.

Here, the noun is birou, which is a neuter singular noun. In Romanian, neuter nouns usually behave like masculine in the singular, so the form is:

• propriul birou

If the noun were feminine, you would use a feminine form:

• propria cameră = her own room
• propria mașină = her own car

So even though ea is feminine, the adjective matches birou, not ea.

Because propriul includes the definite article.

The basic adjective is propriu = own / proper.
When it comes before the noun in this structure, it takes the definite ending:

• propriul birou

So -l here is not random; it is part of the definite form.

Because the definiteness is already being carried by propriul.

In Romanian, when certain adjectives come before the noun, the definite article can appear on the adjective instead of on the noun:

• propriul birou

Compare this with a post-noun structure, where the noun would normally take the article:

• biroul propriu

So in your sentence, Romanian does not need biroul, because the definiteness is already marked in propriul.

Because propriu often means one’s own, and it usually refers back to the subject of the sentence.

So:

• Ea are propriul birou. = She has her own office/desk.
• El are propriul birou. = He has his own office/desk.
• Eu am propriul birou. = I have my own office/desk.

The owner is understood from the subject.

This is different from simply saying:

• biroul ei = her office/desk

That phrase shows possession, but it does not emphasize own as strongly.

Birou can mean both office and desk, depending on context.

So this sentence could mean:

• She has her own office or
• She has her own desk

Romanian uses the same word for both, so you rely on context to know which meaning is intended.

Yes, but the meaning is slightly different.

• Ea are biroul ei. = She has her desk/office.
• Ea are propriul birou. = She has her own desk/office.

biroul ei is a more neutral way to say that it belongs to her.
propriul birou adds emphasis: it suggests something like her own personal, separate, or exclusive office/desk.

So if you want the sense of own, propriul birou is better.

Yes, Romanian can place adjectives after the noun, but in this case propriul birou is the more natural and common way to express her own office/desk.

You may also see:

• biroul propriu

But propriul birou is the clearer and more usual form for learners to remember.

So as a default, Ea are propriul birou is an excellent model sentence.