The issue of journal abbreviations is treated in detail in many posts, but none of them seems to address it in conjunction with exporting to a bibtex file (or, better, to the synchronised bibtex files).
I followed the steps found elsewhere to get the journal abbreviations to be shown in Mendeley if the Citations view is used (http://support.mendeley.com/customer/portal/questions/124632-how-to-enable-journal-abbreviations-), and this works together with a hacked version of a bibtex style (http://support.mendeley.com/customer/portal/questions/243441-customize-journal-abbreviations-and-bibtex-export-style).

In this way, the citations can be successfully 'copied' to an ASCII file, with the correct bibtex syntax and journal abbreviations. However, if they are 'exported' to a Bibtex file, the abbreviations are not used. The same holds in the synchronised bibtex files. Without the journal abbreviations, the produced bibtex files are useless, because the latex output would not follow most journals' guidelines (see e.g. the Astronomy & Astrophysics or Monthly Notices of the Royal Astronomical Society guidelines).

This is a major issue, because it prevents bibtex users to use Mendeley as THE reference organizer. Of course, one could write a simple script that fixes the bibtex files produced by Mendeley, but it would need an external step, while automatic synchronization between Mendeley and bibtex is absolutely at reach.

ubuntu@ubuntu-desktop:~$ mendeleydesktop
Using system Qt version 4.6.2 in /usr/lib
Running /usr/bin/../../opt/mendeleydesktop/bin/mendeleydesktop
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-cif" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-cml" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-daylight-smiles" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-gamess-input" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-gamess-output" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-gaussian-log" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-mopac-out" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-pdb" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-xyz" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-gaussian-checkpoint" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-gaussian-cube" is an invalid MIME type ("chemical" is an unregistered media type)
Error in file "/home/~/.local/share/applications/avogadro.desktop": "chemical/x-qchem-output" is an invalid MIME type ("chemical" is an unregistered media type)
Warning in file "/home/~/.local/share/applications/google-chrome.desktop": usage of MIME type "x-scheme-handler/http" is discouraged (the use of "x-scheme-handler" as media type is strongly discouraged in favor of a subtype of the "application" media type)
Warning in file "/home/~/.local/share/applications/google-chrome.desktop": usage of MIME type "x-scheme-handler/https" is discouraged (the use of "x-scheme-handler" as media type is strongly discouraged in favor of a subtype of the "application" media type)
Warning in file "/home/~/.local/share/applications/google-chrome.desktop": usage of MIME type "x-scheme-handler/ftp" is discouraged (the use of "x-scheme-handler" as media type is strongly discouraged in favor of a subtype of the "application" media type)
Error in file "/home/~/.local/share/applications/sa.edu.ksa.ayat-ayt-ayat.desktop": "ayt/ayat" is an invalid MIME type ("ayt" is an unregistered media type)
Warning in file "/home/~/.local/share/applications/mendeleydesktop.desktop": usage of MIME type "x-scheme-handler/mendeley" is discouraged (the use of "x-scheme-handler" as media type is strongly discouraged in favor of a subtype of the "application" media type)
Error in file "/home/~/.local/share/applications/kde4-kalzium.desktop": "chemical/x-cml" is an invalid MIME type ("chemical" is an unregistered media type)