Mashiach L. Bjorklund
To be honest, there is no guarantee of absolute privacy with a DNA test. All you can do is add levels of anonymity to your test results. Examples: Do not use an identifiable name. Do not attach your DNA results to a tree. Do not make your ethnicity results publicly available. Beyond those simple actions there is not much you can do.
If someone shows up as a DNA relative to me, even with all the above in place, there are enough of my relatives who have taken tests for me use the "genetic distances" of the unknown person to them and me, to allow me to calculate with relative accuracy who they are.
In fact, recently second cousin took a test. They pretty much had done all of the above. Within an hour I was looking at their Facebook profile and pictures of their family. Coincidentally, their Facebook page had a post where they said they had just taken an ancestry DNA test and they posted their ethnicity results.
Probably the best approach is to share with the individual you want to take the test what you have done and discovered already through your test. Perhaps you can convince them to take a leap of faith and try to discover more about your shared family history.