@RFAltersitz Yes...if they click on the icon they can see the information.
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Alerts by definition are intended for information that anyone dealing with the student in the education setting should be aware of. If you want to restrict access to specific users, you might consider setting up a custom page and fields for the information instead.
That said, with custom programming it should be possible to tie visibility to security groups - but the maintenance of that could be easy to lose track of.