Don’t have to be Catholic to have a private or personal intention. Sometimes people will simply ask for prayers, and want to submit to God’s will, but do not make public what the intention is.

From the Catholic point of view, one does not request prayers for anything sinful, nor does one pray to “Change” God’s will. However it is not always needed to mention the intention, as it or they are known only to God and the person/persons who originally requested the prayers.