Your cause of death isn't a personal thing either by the way, it's a matter of public record in most states (and of course word tends to get out with something as emotional as a death)
If there is public record, they are entitled to that and nothing more.
The death person and his/her family have their own right to privacy and it trumps curiosity of collegues.
Again, this isn't some random suicide, these are people in a known high pressure environment already aware this person committed suicide.
The organization can pretend nothing happened (even though everyone already knew), or they can take charge and try and help these people cope with that devastating news and help anyone who might already be in a bad place...
Once you're dead you're dead, that's why causes of death aren't personal... there's no person left.
The organization doesn't even have to frame it in of terms this person's death, just a simple call out that "if anyone is struggling with this loss or needs help come to us".
Anything is better than pretending it didn't happen.
> Once you're dead you're dead, that's why causes of death aren't personal... there's no person left.
The family and actually close friends are left. And people who are about to die deserve to know their privacy will remain private too.
Your teammates are not entitled to know your cause of death. That's a personal thing, but you aren't there any more to choose if it's okay to share.