How do separating service members or recently separated veterans look for job opportunities overseas?

Try indeed.com or http://www.careerstructure.com/search/international-jobs

For those of you that have transitioned under the current GPS/TRS programs, I recommend contacting the TAP program manager that facilitated tour transition classes. There has been great progress in this area and one of the improvements is the screening of possible employers that want access to our veteran work force. As a result, the TAP / TRS managers for each installation find themselves to have very extensive job distribution lists at the ready and all you have to do is ask to be put in the dostro list. I am in Okinawa and the personnel here a CONOS and Overseas distro list.

Start Googling :) Privatemilitary.org, Indeed.com, Securedaspects.com are good job sites to start with. Some companies are AECOM, Fluor, PAE, Dyncorp International, CACI, Lockheed, Raytheon, Parsons, Booze Allen Hamilton among a multitude of others.
Good luck!