According to the University of Bristol,
A comma is used to set off parenthetical elements in a sentence. The parenthetical element (also known as an aside) is part of the sentence that can be removed without changing the essential meaning of the sentence.
Sarah, the most intelligent pupil in the class, was always late for school.
The pyramids, one of the wonders of the ancient world, lie just outside Cairo.
If you are using a comma to do this, it is important that the aside is opened and closed with a comma. A common mistake is to omit the second comma.
Source: http://www.bristol.ac.uk/arts/exercises/grammar/grammar_tutorial/page_06.htm
Applying that 'test' to your pair of example sentences,
The CSI, by its email, has informed us that ...
The CSI has, by its email, informed us that ...
you can see that the part within comma parenthesis, "by its email", is the same in both cases and you can remove that part without changing the essential meaning of both remaining sentences, which are also identical: "the CSI has informed us that..." Only the detail "by email" is lost and the basic meaning remains. That shows that comma parentheses have been correctly applied in both sentences.
You can even place the phrase within comma parentheses elsewhere in the sentence, as in
The CSI has informed us, by its email, that ...
Again, removing that parenthetic phrase gives the same basic sentence as the earlier examples: "the CSI has informed us that..."
In short, only the position of the phrase within comma parentheses varies in your examples, in grammatically legitimate ways without changing the meaning, and both your sentences are equally correct.
Related (complex) question asked previously here, for your further reference: Parenthetical commas
Note: Comma parentheses are governed by much the same general rules as other types of parentheses and can be replaced by a pair of round brackets or a pair of dashes in most constructions, but whether you use commas, dashes or round brackets, the parenthetic phrase is not to be placed in 'odd locations' such as between "the" and "CSI", or between "informed" and "us", as rightly pointed out in comments by @Edwin Ashworth, because that would create ungrammatical sentences such as
the, by its email, CSI has informed us that...
the CSI has informed (by its email) us that...
It seems logical to me to suggest that placement of the parenthetic phrase would be governed by the condition that the sentence should make general grammatical sense as a whole if the parenthetic punctuation (but not the parenthetic material) is removed, as in
the CSI has by its email informed us that...
the CSI has informed us by its email that...
but not
the by its email CSI has informed us that...
the CSI has informed by its email us that...
I am still looking for an authoritative rule on where and where not to place parentheses but am yet to find it because many online resources dealing with parentheses seem less concerned with grammatically correct sentence structure than with punctuation, and appear content to say some version of "the parenthetic statement should be removable without changing the essential meaning of the sentence", possibly assuming that where to place it is a matter of common sense for a reasonably proficient user, and the reader already knows where to correctly place the parenthetic statement? I shall post the appropriate reference for placement as soon as I find it.
