Same review for Math Reviews and Zentralblatt I have been serving for several years as a reviewer for both Mathematical Reviews (MathSciNet) and ZbMath.

Earlier this year, I received an invitation for a paper for MR which I already reviewed for ZbMath. I dug up my old review and submitted it to MathSciNet without giving it much thought.

But there seems to be a pattern emerging. I received an invitation from MR for two more papers I already reviewed for ZbMath.

So here are my questions:


Do the editors of MR know/check who reviewed the paper for ZbMath in order
to find suitable reviewers?
What do I do about the double reviews? 


My idea was to just refuse to write the reviews. But if the answer to the first question would be yes, then the editors know that I have a thorough review lying around. Just resubmitting this one would save a colleague from writing a new one.
The Math Reviews Guide for Reviewers addresses this explicitly:


  Reviews in other publications. You may find yourself asked to write a review of the same item for another publication. In general, the community is best served by reviews written independently. If you have already been asked by another publication to review an item, please let us know and we will transfer it to another reviewer.


So, if you have already reviewed the article for Zbmath, you should decline to review it for MR.  I didn't find a similar rule on the Zbmath side, but I would say the converse should also hold.

Also, per your agreement with Zbmath, when you submit a review to them, you grant them "exclusive exploitation rights".  So if you submit the same review to MR, you're infringing Zbmath's copyright.  I doubt they will sue you over it, but this shows it's actually illegal as well as unwanted.

In view of this, you may want to contact MR and ask them to withdraw your previously submitted review.

Evidently, one should not assume that the MR editors will check who reviewed the article for Zbmath in order to avoid inviting the same person, since they apparently failed to do so in your case.  (Another possibility is that they did check, but your Zbmath review hadn't been posted yet.)