The coins given by a father to a Kohen for the Misva of Pidyon Ha’ben must belong to the father. It goes without saying that a father cannot fulfill this Misva with money that was obtained unlawfully.
The father must actually give the coins to the Kohen. If the Kohen tells the father that he is "Mohel" – meaning, he foregoes on the money – then the Misva is not fulfilled. The father must give the Kohen the money. The Kohen may return the money to the father if he so wishes – and the Gemara relates that this was the practice of two great sages who were Kohanim – but the Misva is not fulfilled unless the Kohen actually receives the coins from the father.
As the Pidyon Ha’ben requirement constitutes a financial obligation, we apply to this requirement the famous rule of "Ha’mosi Me’habero Alav Ha’re’aya" – that the burden of proof falls upon the one seeking to extract money. This means that in a case where there is some uncertainty as to whether the obligation of Pidyon Ha’ben applies, the father is not required to pay money to a Kohen. Unless the Kohen can prove that the father owes the money, the father is not required to pay. One case would be if it is unknown whether the father himself is a Kohen, or if the mother is a Kohenet. If either parent is a Kohen or Kohenet, the obligation does not apply. Therefore, in a situation where it is unknown whether the father is a Kohen or the mother is a Kohenet, the father does not have to pay Pidyon Ha’ben until it can be proven that he is not a Kohen and the mother is not a Kohenet.
Summary: The money paid to a Kohen for Pidyon Ha’ben must legally belong to the father, and must actually be given to the Kohen; the Misva is not fulfilled if the Kohen says he foregoes on the payment. In a situation where there is some uncertainty as to whether the Pidyon Ha’ben obligation applies, the father is exempt until it can be ascertained that the Misva applies in that case.