To share incidents publicly, you will first need to allow sharing of incidents for your entire organisation globally.
Head over to organisation settings and select your choice to enable or disable your team from sharing incidents publicly.
To share an incident, you will need to click on the Share button on the right-hand side of the incident details page.
The link for the public page will not be unique for each time to decide to share or not. However, the public page link is unique to each incident.