Will NPS survey responses still be recorded from anonymous users even if my JIRA is setup privately?
No. NPS survey responses will be recorded, even if JIRA does not allow anonymous login. The only thing that's necessary is that the reporter users configured in the survey still has the right to create issues.
You do need to ensure as well that your JIRA instance is accessible on the public internet (or from whichever network your users are accessing the site that you embedded the survey on).
You must be a registered user to add a comment. If you've already registered, sign in. Otherwise, register and sign in.