I think it just as important to control quality as to contain costs, so I would recommend that the federal government require all private contractors to meet a set of health, safety, and environment care standards similar to its own internal standards--whether this be in the United States or overseas.
After working as a contract wildland firefighter and as a Americorps forestry volunteer (separately), I have seen very different sets of standards being enforced with private contractors, compared to agency standards. As a somewhat worrying trend, government agencies under the Bush Administration sought to increasing privatize government services that the US Forest Service or Bureau of Land Management typically carried out. These private contractors had very different standards (much lower, generally) than the federal government and frequently violated work and safety standards. For example, wildland firefighters are increasingly at risk due to poor rest to work ratios, that can result in poor judgment, work injuries, and loss of life in some instances due to lack of sleep. I have seen firefighters falling asleep on the fireline and experienced dehydration/heat exhaustion due to improper access to drinking water in high temperatures (100F+) in full sun. While this particular fire was on privately-owned land, I have heard similar accounts of poor working conditions on small-medium federal fires, including sexual harassment. In federal agencies, such work conditions would never be permitted, and safety checks are regularly performed.
I think the federal government should require all bidding contractors to sign on to best practice federal work policies, BEFORE the start of the bidding process and agree to monitoring by a federal representative.