“Numbers” or “number?” Which sentence is correct?

The short answer is that it should be in the singular. (I will explain this later.)

However, there is a better solution to this—one that is both syntactically correct and which sounds more natural:

In 2000, Vietnamese students who studied in Russia and France numbered around 3 million and 3.5 million, respectively.

This dispenses with the troublesome noun number, using instead the past tense of the verb number.

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To explain why the example sentence is in the singular, consider its expanded version:

In 2000, the number of Vietnamese students who studied in Russia was around 3 million and the number of Vietnamese students who studied in France was around 3.5 million.

In both clauses, number is singular and so is the verb. Shortening it into a single clause with respectively doesn't change this essential fact in this particular case. (Even when X is plural, you can't say the numbers of X.)

In 2000, the number of Vietnamese students who studied in Russia and France was around 3 million and 3.5 million, respectively.

But in addition to the rephrasing I suggested at the start of my answer, there is another possibility:

In 2000, the total number of Vietnamese students who studied in Russia and France was around 6.5 million: 3 million and 3.5 million in the respective countries.