One of the few methods for detecting a large proportion of any program's is to allow many people, with all their varied perspectives and skills, to examine it, by proclaiming that the program is free and open source and releasing both the source code and binaries for inspection.

That's a claim often made ("With enough eyes all bugs are shallow") but it's not so clear-cut in practice. In real life a lot of open-source projects are very buggy and remain very buggy (and open to 'sploits) for a very long time. At the same time there is closed-source software which is considerably more bug-free (but very expensive) -- e.g. the code in fly-by-wire airplanes.

Besides, physical control, generally speaking, trumps all. If your mind is running on top of, say, open-source Ubuntu 179.5 Zooming Zazzle but I have access to your computing substrate, that is, the physical machine which runs the code, the fact that the machine runs an open-source OS is quite irrelevant. You're looking for impossible guarantees.

And remember, that you are not making choices, but requests. You can't "trust the motives" or not -- if someone revives you with malicious intent, he can ignore your requests easily enough.