genetic testing, insurance coverage, privacy issues?

Hello, I am new to the forum. I just turned 36. I had a third early
miscarriage 2 weeks ago. We are now considering a more serious work up.
Our genetic counselor suggested we do chromosome blood routine for both
of us. I believe our insurance will cover it but she mentioned that some
people opt out and pay for the tests on their own for privacy issues.Has
anyone heard of or done this? What privacy issues could be there? For
coverage of any possible  treatments in the future, is it better to pay
on your own or go through the insurance? The tests are expensive so if
anyone can  shed any light on the topic I would be very grateful.

Many thanks,

miva