If someone records my Skype conversation, is that video his property or mine?
Is there a law against him making that video public or even recording it to begin with?
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As stated in the answer to What is considered "public" in the context of taking videos or audio recordings?; if either of the participants is in Australia than unless all parties have given consent then the recording is illegal.
Notwithstanding its legality, property in the recording vests in the person who made it. There is no law against him keeping it. There is no law against him publishing it unless the material contained is offensive, hate speech or defamatory (see Customer feedback gathering in Australia).