I wrote a post ages ago about some of the politics of doing field oversight on another environmental firm - how, even in a situation where your clients may be at odds, you generally work fine together.
Performing field oversight is generally a job left to lower-level field people. It's not terribly taxing, and the field representative who's doing the technical work generally knows what to do. In fact, it's often used as training.
After I wrote that post, I found myself in a new situation - I was the person to be overseen, and the person there to keep tabs on me had never seen this stuff before. I'm never one to let a teachable moment go, so I was happy to explain how everything worked, mechanically, and if there were any issues with the work (we were drilling, so...yes, there were issues), I pulled the person to the side to explain what had happened and suggest alternate methods the driller could use to avoid that problem. For example, why was it taking so gosh-darn long to drill this particular borehole? Well, probably the drill bit was worn out back when we were going through an especially nasty section earlier, but the driller wanted to fully use up the bit (and didn't want to yank up all the rods attached to said bit if he didn't have to).
It was a little strange to be training someone who was supposed to be keeping tabs on me. After all, this newbie could shut us down and kick me off the site if I wasn't fully cooperative. What helped was honesty. I told the oversight person, "Look, I'm not perfect. We are trying to do this job as safely and technically correct as possible, but if you see anything that you're not comfortable with, please let me know right away and we will address it, whether that means fixing something right now and continuing work, or calling our respective bosses to figure out a better way to do things."
Because the oversight person was comfortable coming to me with questions, I was able to explain stuff that was new/different to the newbie, we had an extra pair of eyes keeping an eye on stuff that may leak or break or get tripped over, and I didn't have high-level folks from the oversight contractor on my ass about every field decision.