Problem

Comment restriction was added to the new issue view, but it didn't include team-managed (formerly next-gen) projects, so currently, the comments are visible to anyone that has access to the project.

Suggestion

Add the comment restriction feature to team-managed (formerly next-gen) projects.

Workaround

The comment visibility can be set from the search results in the Issue Navigator (List view).

1. Search for issues in the Issue Navigator
2. Switch to List view
3. In the list view, you can choose to add comment and perform other actions on individual issues without opening them
4. Add comment and notice you'll have an option to restrict the visibility

Note: Once the comment visibility is set it can't be edited from the issue view.

Comment visibility can be removed using the Update comment REST API call:

curl --request PUT \
  --url 'https://your-domain.atlassian.net/rest/api/2/issue/{issueIdOrKey}/comment/{id}' \
  --user 'email@example.com:<api_token>' \
  --header 'Accept: application/json' \
  --header 'Content-Type: application/json' \
  --data '{
  "body": "Remove comment visibility restriction.",
  "visibility": null
}'

Also, while trying to set the visibility using the workaround, you may notice that not all groups and roles are listed. Please make sure that you are part of the group or role you wish to restrict the comment to.
Otherwise, you will not be able to see it in the drop-down list.