A joint petition by definition is a settlement. A standard joint petition will have both parties waive the right to notice of entry of the Decree, among other waivers. If you use the forms from the Family Court Self Help Center, that language is included.
So, you will generally not have to "serve" the other side with anything.
If there is a litigated divorce you need to have the other party properly served. If it's being done on consent then signed documents will suffice.
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There is no service of process with a joint petition. You both have to sign, and then file all the necessary papers with the court to be processed. These "do it yourself" projects have a high failure rate in the court system. It is worth consulting with an attorney to make sure you are doing it right.
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