What VOR said there, SKJohn.
If you've seen pictures of coalition troops headed out of the wire with AK's, my guess they were probably actually: 1 - Private Military Contractors, that yes, do buy local AK's for their job. 2 - Operators for Spec Ops, but then, you should not be seeing pics of them at all. 3 - Detached soldiers or operators working and training alongside Iraqi soldiers (MiTT's or SF Green Berets for example), which then often use the same weapons and rounds that the Iraqi's use. Easier to operate if you don't have many weapons with many types of ammo requirements for small units.
AK's are taken as trophies in country by soldiers, but most often do not make it back stateside, unless as part of a Company's or Battalion's trophy for display.
Now, there have also at times (OIF-II) been ammo shortages experienced by troops (often National Guard or Reserve troops it seems....grrrrr....) in country, where sometimes, illegally, and unofficially, said troops will augment their issued weapons with local weapons and munitions (Russian PKM and over 1,000 rounds). But then, if said troops could have gotten their weapons' basic load requirements and resupply in the first place...... this would not have occurred at all. By OIF-5, this should not be an issue any more.
But, in general, it is not the practice for common soldiers in country to make use of non-issue (but not necessarily un-modified) weapons.