Yes, the critters taught us how to build our own. Some have been seen with NASA markings and some with USAF markings.
Certainly AirForce personnel have been seen working around some of the craft and on some of the critters mother ship craft.
There IS, however, a BIG DIFFERENCE . . . in our crews and craft and THEIR crews and craft.
By Them and Theirs, I assume you mean NASA. If you mean USAF as “we”. About the only aliens on post would be named “Juan” and “Pedro”.
Personally, I thought it was a hoot when that Air Force bird stripped its gear shift over Phoenix. About the only flight path from Nevada heads East to Flagstaff, then due South pretty much flying the Verde River, then through Phoenix and West again to San Diego. This is because off the coast there is about the only place in the US you can go supersonic without getting everybody pissed off.
Well, about half an hour before sundown, that poor pilot got stuck in hover mode, above two hundred feet above some upscale homes, and had to wait until the breeze pushed him out over West Phoenix desert, so the repair guys could fix him up.
And the cover up was a hoot. Such a big deal about a stealth aircraft with quiet engines that can hover. Even an ex-governor was told to put a sock in it. Yeah, yeah, national security.
The local news was funny. First they showed a horizon view of the city at night, from about 20 miles away. “Nope! No UFO there!” Then they went into extended harangues, claiming “UFO nuts” were making “outrageous claims”. I’m surprised that they didn’t show Loch Ness monster and bigfoot pictures, while saying “And these are other things the nuts believe in! Anyone who says there was a UFO is *just* *like* *them*!”