Can a petitioner personally give a legal court form to the respondenent?

I am assuming this is a document requiring service pursuant to Arizona Rules of Family Law Procedure 43. If so, you may not hand-deliver the papers to the Respondent unless he or she is willing to sign an Acceptance of Service in front of a Notary or Clerk of the Superior Court and return it to you to file with the court (or file it himself).  If you are in Maricopa County, you can get the forms you need from their web site at clerkofcourt.maricopa.gov.

In the alternative, you may want to consider service by certified mail return receipt (with affidavit), service by process server, or service by sheriff.