Beide zijn nog in de bibliotheek.

Yes. Beide means both.

In this sentence, beide is being used on its own, almost like a pronoun: both (of them).

So:

• Beide zijn nog in de bibliotheek. = Both are still in the library.

You could imagine an understood they in English, but Dutch does not need to say it here.

A closely related form is:

• beide boeken = both books

There, beide directly modifies a noun.

Because beide refers to two people or things, so the verb must be plural.

• is = singular, from zijn = to be
• zijn = plural

Compare:

• Hij is in de bibliotheek. = He is in the library.
• Beide zijn in de bibliotheek. = Both are in the library.

Since both means more than one, Dutch uses the plural verb zijn.

Nog here means still.

So the sentence suggests that the two people are still in the library, perhaps longer than expected.

• Beide zijn in de bibliotheek. = Both are in the library.
• Beide zijn nog in de bibliotheek. = Both are still in the library.

Nog is a very common Dutch word, and its meaning depends on context. It can mean things like:

• still
• yet
• another / one more
• again in some contexts

But in this sentence, still is the natural meaning.

Because in normal Dutch word order, short adverbs like nog often come before a prepositional phrase such as in de bibliotheek.

So this order is natural:

• Beide zijn nog in de bibliotheek.

It would sound less natural to an English learner to move things around too freely. Dutch word order is often stricter than English in this kind of sentence.

A useful pattern is:

• subject
  • verb
    • adverb
      • place

For example:

• Hij is nog thuis.
• Zij zijn nog op school.
• Beide zijn nog in de bibliotheek.

Because in de bibliotheek describes a location: they are in the library.

• in de bibliotheek = in the library
• naar de bibliotheek = to the library

So:

• Beide zijn nog in de bibliotheek. = They are there now.
• Beide gaan naar de bibliotheek. = They are going there.

English also makes this distinction, but Dutch learners often need to watch for it carefully.

Because bibliotheek is a de-word in Dutch.

Dutch nouns usually take either:

• de
• het

You generally have to learn this with each noun.

So:

• de bibliotheek = the library

There is not always a simple rule that lets you predict whether a noun takes de or het, though there are patterns. For this word, you simply learn it as de bibliotheek.

Yes, often you can.

• Beide zijn nog in de bibliotheek.
• Allebei zijn nog in de bibliotheek.

Both can mean both are still in the library.

However, there is a slight difference in feel:

• beide can sound a bit more neutral or slightly more formal/written
• allebei is very common in everyday spoken Dutch

So in casual speech, many speakers would naturally say:

• Allebei zijn nog in de bibliotheek.

But beide is completely correct and common too.

It can refer to both people and things.

Examples:

• Beide zijn nog in de bibliotheek. = both people are still in the library
• Beide boeken liggen op tafel. = both books are lying on the table

So beide simply refers to two items, whether they are people, objects, ideas, and so on.

The exact meaning depends on the context.

Yes, absolutely. That is a very natural sentence.

• Ze zijn nog allebei in de bibliotheek.

This makes the subject ze explicit: they are both still in the library.

Compared with Beide zijn nog in de bibliotheek, this version may sound a bit more conversational because English speakers are often used to having an explicit subject like they.

So both are fine, but they are structured differently:

• Beide zijn nog in de bibliotheek. = Both are still in the library.
• Ze zijn nog allebei in de bibliotheek. = They are both still in the library.

It is mostly neutral.

• beide gives it a slightly neat or written feel
• but the sentence is not especially formal

In everyday speech, many people might choose:

• Ze zijn nog allebei in de bibliotheek.

That sounds a bit more conversational.

Still, Beide zijn nog in de bibliotheek. is perfectly natural Dutch and not overly formal.

A rough pronunciation guide is:

• Beide ≈ BY-duh
• zijn ≈ a sound somewhat like zine, but with the Dutch ij, which does not match English exactly
• nog ≈ nokh or nogh with a guttural Dutch g
• in de bibliotheek ≈ in duh bib-lee-oh-TAYK

A rough full version:

• BY-duh zayn nokh in duh bib-lee-oh-TAYK

A few notes:

• The Dutch ij in zijn is tricky for English speakers.
• The g in nog is the Dutch guttural sound, not the English g in go.
• bibliotheek has the main stress on the last syllable: -theek.

If you want to sound natural, focus especially on zijn and nog, since those contain sounds English speakers often find difficult.