How can I expose CiviPetition signatures to the public?
This isn't a function out of the box. If you are in drupal, you can use view to display the list of activities (a petition signature is "simply" an activity of type "petition signature". They are others ways to fetch data out of civi on other CMSes, but might need to write some code.

As a side note: please don't display the full name of the signatories, that might be a risk for them. It happened not so long ago that a right wing group took a list of a progressive petition and started to dox them (find their address and contact details...), contact them...

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