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What is "Acceptance of Service" by the other person?

This is a form provided by the court clerk, that the other person has to sign in front of a notary public or clerk of the court, saying that they got a copy of the papers you filed.  By signing they are not agreeing with what the papers say, just that they received them.  They also admit that they got them without having to be served in person by a sheriff or process server.

Resources:

Maricopa Acceptance of Service Form