The Army trains most of their own dogs. Some programs, like the SF and TEDD programs are civilian trained for the most part.
Green dogs are trained the basics of detection and bite work by experienced trainers and then placed with a handler where they continue to train and learn as a team. As a handler, you may get a green dog, or a dog that has had previous handlers. In theory, all the dogs are trained the same and can easily switch between handlers. The handlers are far harder to train than the dogs. Less than 40% of my class actually certified with a dog.
It is important for personalities to match up between handler and dog, especially with new handlers. When pairing dogs and handlers, trainers try and make as good a match as they can, depending on how many dogs they have available.
It is possible for a dog to form a great bond with a new handler. Fama was 4 when I got her, and I was her 5th handler. I am the first one she spent more then a few months with, as she very much enjoyed punking out new handlers and sending them to the emergency room for stitches. We formed as tight a bond as I can imagine over the course of several months.
I hope this helps clear some things up.
When a dog saves the life of a man, it becomes clear that partnership knows no bounds.
Fama - T.E.D.D. OEF XI-XII (GSD)(RET)
Marshall - T.E.D.D. OEF XII-XIII (Lab)(SF EDD)
Lucian - Med Alert / Mobility SD (Cane Corso)
Pud - the old man (Pit x Lab)