Also, im not sure its a great idea for your competition to be able to see what bugs you are tracking/fixing.
Nonsense.
Being more open is far better then keeping tight lipped.
Look at GPUOpen, once they made that online and open to all, look at all the good that has done.
AMD has also released a ton of open source code, that also isn't hurting them.
I already mentioned MS, and they are not suffering because everyone can see what bugs are out there.
Look at Linux, they got open bug tracks, same with Qt, and a ton of other companies / packages do as well.
If it is security related, then sure, that is why they have private flags, don't need to advertise those.
Frankly, it would be another win for AMD being more open, they already got forums where people discuss bugs anyway, but forums are very bad for bug reports, this would just be a extension of that to get better reports to drill down to the issue at hand.
Jira is pretty nice, have used it myself in the past.