معجون الأسنان قرب خيط الأسنان.

In Arabic, the present-tense to be is usually not stated in simple sentences like this. So:

معجون الأسنان قرب خيط الأسنان

literally looks like toothpaste near dental floss, but it means the toothpaste is near the dental floss.

If you wanted a past-tense version, then Arabic would use a verb such as كان.

It is a nominal sentence in Arabic.

A simple way to break it down is:

• معجون الأسنان = the topic / subject
• قرب خيط الأسنان = the comment / predicate

So the structure is basically:

[The toothpaste] [near the dental floss]

Here قرب means near or in the vicinity of.

Grammatically, it is often treated as an adverbial expression of place, and it is followed directly by the noun it relates to:

• قرب خيط الأسنان = near the dental floss

So this is a normal Arabic pattern. You do not need من after قرب in this structure.

Because Arabic is using two separate iḍāfa constructions:

• معجون الأسنان = toothpaste
• خيط الأسنان = dental floss

Each expression literally means something like:

• معجون الأسنان = paste of the teeth
• خيط الأسنان = thread/floss of the teeth

Since both items are related to teeth, الأسنان appears in both phrases.

That is just a difference between how the two languages build these expressions.

English often uses a singular noun in compounds:

• toothpaste
• toothbrush

Arabic commonly uses الأسنان = the teeth in these set expressions:

• معجون الأسنان
• فرشاة الأسنان
• خيط الأسنان

So even though English says tooth-, Arabic naturally says of the teeth.

Because معجون الأسنان is an iḍāfa construction.

In an iḍāfa:

• the first noun usually does not take الـ
• the whole phrase becomes definite because of the second noun

So:

• معجون الأسنان = the toothpaste
• not المعجون الأسنان

The definiteness comes from الأسنان.

The same applies to:

• خيط الأسنان = the dental floss

Grammatically, it is singular, because خيط is singular.

So خيط الأسنان is a singular phrase meaning dental floss as a product or item.

Even if dental floss consists of many fibers or could be thought of as a substance, Arabic commonly names it with singular خيط.

In fully vowelled formal Arabic, it would be:

مَعْجُونُ الأَسْنَانِ قُرْبَ خَيْطِ الأَسْنَانِ

A grammatical breakdown:

• مَعْجُونُ: nominative, because it is the subject/topic
• الأَسْنَانِ: genitive, because it is the second part of the iḍāfa
• قُرْبَ: accusative as an adverbial expression of place
• خَيْطِ: genitive, because it follows قرب in this construction
• الأَسْنَانِ: genitive again, because it is part of the second iḍāfa

In normal everyday writing, these short vowels are usually omitted.

Yes. That is also correct and very natural.

There is a small structural difference:

• قرب خيط الأسنان = near the dental floss
• قريب من خيط الأسنان = close to the dental floss

So:

• قرب is a compact location expression
• قريب من uses the adjective قريب plus the preposition من

Both are good Arabic. The original sentence sounds especially neat and compact, like something in a textbook, exercise, or simple location description.

Because قرب and قريب behave differently.

• قرب + noun
  قرب خيط الأسنان
• قريب من + noun
  قريب من خيط الأسنان

So if you use قرب, you can directly attach the following noun.
If you use قريب, you usually add من.

That is why the original sentence is correct as it stands.

A simple pronunciation guide is:

maʿjūn al-asnān qurb khayṭ al-asnān

If you want a more careful reading with case endings, it is closer to:

maʿjūnu l-asnāni qurba khayṭi l-asnāni

A few sounds may feel unfamiliar to English speakers:

• ع in معجون: a deep throat sound
• خ in خيط: like the ch in Scottish loch or German Bach
• ط in خيط: a heavy/emphatic t

It is correct Arabic, but it does sound a bit like a simple descriptive or classroom sentence, especially because it is so short and direct.

A native speaker might also say things like:

• معجون الأسنان بجانب خيط الأسنان
• معجون الأسنان قريب من خيط الأسنان
• يوجد معجون الأسنان قرب خيط الأسنان

But the original sentence is perfectly good for learning basic vocabulary and sentence structure.