Biblioteket er et offentligt sted, men denne besked er privat og skal ikke læses højt.

Because Danish often puts the definite article at the end of the noun instead of using a separate word like the.

• et bibliotek = a library
• biblioteket = the library

So biblioteket is the definite form of bibliotek. This tells you the sentence means the library, not just a library.

Because sted is a neuter noun in Danish, so it takes et in the indefinite singular.

• en is for common-gender nouns
• et is for neuter nouns

Since sted is neuter, you get:

• et sted = a place
• et offentligt sted = a public place

The adjective also has to match the noun, which is why offentlig becomes offentligt here.

Danish adjectives usually add -t when they describe a singular neuter noun.

Here:

• offentlig = public
• et sted = a place, with sted being neuter
• so: et offentligt sted

Compare:

• en offentlig bygning = a public building
• et offentligt sted = a public place

So the -t is an agreement ending.

Yes. Both can mean this message.

• denne besked is correct and a bit more formal or written
• den her besked is very common in everyday spoken Danish

Also, besked is a common-gender noun, so the singular demonstrative is denne. For a neuter noun, it would be dette.

Examples:

• denne besked = this message
• dette brev = this letter, if the noun is neuter

Because privat is agreeing with besked, and besked is singular common gender.

In Danish adjective agreement:

• common-gender singular: usually basic form
• neuter singular: often -t
• plural or definite forms: often -e

So:

• en besked er privat = a message is private
• et brev er privat or sometimes privat as a fixed form in predicate use, depending on adjective pattern
• beskederne er private = the messages are private

In this sentence, privat is the correct singular form after er.

They are doing two different jobs.

• offentligt is directly describing the noun sted
• privat comes after er, so it is a predicate adjective

This is similar to English:

• a public place
• the message is private

In Danish:

• et offentligt sted
• beskeden er privat

So the position changes depending on whether the adjective comes inside the noun phrase or after a linking verb such as er.

Because the subject is still denne besked for both verbs.

The structure is basically:

• denne besked er privat
• denne besked skal ikke læses højt

Since the subject is the same, Danish often leaves it out in the second part:

• denne besked er privat og skal ikke læses højt

English does the same:

• This message is private and should not be read aloud

You could repeat the subject for emphasis, but it would sound heavier.

It is a passive construction with a modal verb.

• skal = shall, should, is to, must, depending on context
• læses = be read, in passive form
• skal ikke læses = should not be read, must not be read, is not to be read

In this sentence, it sounds like an instruction or rule: the message is private, so it is not supposed to be read aloud.

Danish often uses -s to form the passive.

• læse = to read
• læses = be read, is read

So:

• Bogen læses = the book is read
• Bogen skal læses = the book is to be read

In your sentence:

• skal ikke læses højt = should not be read aloud

Danish could also use a form with blive:

• skal ikke blive læst højt

But the -s passive is very common and compact, especially in written instructions.

Because ikke usually comes after the finite verb and before any infinitive or participle.

Here the finite verb is skal, so the order is:

• skal ikke læses

That is normal Danish word order.

Compare:

• Beskeden skal ikke læses højt
• Beskeden læses ikke højt

In the first sentence, skal is the finite verb, so ikke comes after skal.
In the second, læses is the finite verb, so ikke comes after læses.

Here højt means aloud or loudly.

The phrase læse højt is a common Danish expression meaning to read aloud.

It comes from the adjective høj, which can mean high or loud depending on context. When used adverbially, it often takes -t:

• en høj lyd = a loud sound
• at tale højt = to speak loudly
• at læse højt = to read aloud

So højt is functioning like an adverb here.

Because men links two main clauses, and Danish normally uses a comma there.

The two parts are:

• Biblioteket er et offentligt sted
• denne besked er privat og skal ikke læses højt

So the comma helps separate two full statements joined by men.

This is very similar to English punctuation in a sentence like:

• The library is a public place, but this message is private.