I'm not sure exactly. If they put you in the Lear, it's totally legit because that aircraft requres two pilots. In the bonanza or caravan you really can't log it unless they have a two crewmember requirement in their certificate, which I'm pretty sure they don't. You could also log PIC if you're sole manipulator on flights that aren't operated 135 (IE reposition flights).
I had some questions for them and sent an email about a year and a half ago. They were answered pretty fast. I'd suggest doing that. Airnet is a good company, and one of the great things about this program is that you're paid. Many companies throw people in the right seat and charge them.