Publishing is the start of the process rather than the end of it. Here is what actually happens, in order.
Review
New petitions are checked against the community guidelines before they appear publicly. Most are approved. The usual reasons for rejection are naming individuals as criminals, publishing someone's personal details, or an ask so vague that no authority could act on it.
Gathering signatures
Once live, anyone with a verified account can endorse it, once. Signatures show a username, never a real name, and the list is never sold or handed to the authority as personal data.
The first endorsement makes it permanent
The moment somebody signs, the petition can no longer be edited or withdrawn by the person who started it. Those people put their names to that particular wording, and changing it afterwards would misrepresent what they backed. Deleting it would take their signatures with it.
So check it carefully before you share it. Until the first endorsement arrives you can still change anything or remove it entirely.
You can always post updates, which add to the record instead of altering it. Moderators keep the power to remove a petition that breaks the guidelines, however many people have signed.
Discussion
Each petition has a discussion board. Comments are moderated, and as the person who started the petition you can mute an account from your own petition's discussion if someone is disrupting it.
Updates and closing
You can post updates at any time, and signatories who opted in are notified. Petitions run for a set period and then close to new signatures, remaining visible as a record.
A closed petition is still useful. It is dated, public evidence that a request was made and how many people backed it, which matters if the same problem has to be raised again.
Common questions
How long does review take?
Usually within a working day.
Can I delete my petition?
Only before it has been endorsed. Once someone has signed it, the petition belongs to everyone who backed it and deleting it would erase their signatures too. Moderators can still remove a petition that breaks the guidelines.
What happens to my petition when it closes?
It stays online as a record, with its signature count and discussion intact. It simply stops accepting new endorsements.