No Basenji is ever totally trustworthy off leash, their prey drive and natural instincts are too high. However depending on the dog and how training has gone, it can be a much smaller risk. But it's still a risk. I let my dogs off leash a lot, and I have put a ton of work into them, but it is a chance I have to take.
And yes, they definitely respond different to their "person" over anyone else. My dogs are very well trained, but when my family tries to give them the same commands, the dogs just come and look at me like "who the heck are they to tell me something?" lol And when my family had led the dog out in the neighborhood he wouldn't come to my dad, I had to go running down there and call him. He was busy checking out the dogs in the yards.