What is the bleu score of professional human translators?
The original paper "BLEU: a Method for Automatic Evaluation of Machine Translation" contains a couple of numbers on this:


  The BLEU metric ranges from 0 to 1. Few translations will attain a score of 1 unless they are identical to a reference translation. For this reason, even
  a human translator will not necessarily score 1. It
  is important to note that the more reference translations per sentence there are, the higher the score is. Thus, one must be cautious making even “rough”
  comparisons on evaluations with different numbers
  of reference translations: on a test corpus of about
  500 sentences (40 general news stories), a human
  translator scored 0.3468 against four references and
  scored 0.2571 against two references.


But as their table 1 (providing the numbers compared to two references, H2 is the one mentioned in the text above) shows there is variance among human BLEU scores:



Unfortunately, the paper does not qualify the skill level of the translators.