Gemara

Clearly, the father blesses Asher Kidshanu b'Mitzvosav v'Tzivanu Al Pidyon ha'Ben (it is his Mitzvah).

Question: Does the Kohen (who receives the money) bless Shehecheyanu, or does the father bless?

Perhaps the Kohen blesses, for he benefits;

Or, perhaps the father blesses, for he does a Mitzvah!

Rashash (DH Kohen): It is clear that either could bless. The question is, who should bless and be Motzi the other.

Answer: The father blesses both.

The Halachah is, the father blesses both.

7a: Rav Papi and Rav Papa argue about whether we bless 'Leva'er Chametz' or 'Al Bi'ur Chametz'. All agree that Leva'er connotes the future; they argue about Al Bi'ur: Rav Papi holds that it connotes the past; Rav Papa holds that it connotes the future.

Question (Beraisa): One who circumcises blesses 'Al ha'Milah' (it connotes the future)!

Answer: There, one cannot bless 'Lamul', for this implies that he himself must circumcise - but really, someone else could circumcise!

Objection: When a father circumcises his son, he should bless 'Lamul' (it is his Mitzvah)!

Answer: Indeed, Rav Papi would say that when a father circumcises, he blesses 'Lamul'!

Rishonim

Rif and Rosh (10:39) cite the Gemara verbatim.

Rambam (Hilchos Bikurim 11:5): One who redeems his son blesses Asher Kidshanu...Al Pidyon ha'Ben, and then Shehecheyanu; then he gives five Sela'im to the Kohen. One who redeems himself blesses Lifdos ha'Bechor and Shehecheyanu.

Rivash Teshuvah 131 (brought in Be'er ha'Golah YD 305:17): Even though the Rambam holds that one blesses with a 'Lamed' for Mitzvos incumbent on himself, Pidyon ha'Ben is primarily on the son himself, just he cannot redeem himself at the time.

Gra (YD 305:15): Our Gemara only discusses a father who redeems. The Rambam says that one who redeems himself blesses Lifdos to resolve our Gemara with circumcision. According to Rashi (7b v'Hilchesa) the Gemara concludes that we always bless 'Al' on circumcision, and the same applies to Pidyon Bechor; we need not make the dubious distinction of the Rivash.

Question: Why do we bless Shehecheyanu for Pidyon ha'Ben, but not for circumcision (unless the father himself circumcises? Nowadays, the custom in Eretz Yisrael is to bless Shehecheyanu in any case.)

Answer #1: We are unsure whether or not the baby is a Nefel.

Rejection #1 (Rosh Bechoros 8:8): We follow the majority of babies, which are viable! Also, in any case it is a Mitzvah!

Rejection #2 (Rashba, brought in Ma'adanei Yom Tov 60): Shehecheyanu does not depend on Simchah, rather, on receiving something useful, even if it is mixed with pain. One blesses Shehecheyanu for an inheritance!

Answer #2 (Rosh ibid.): We do not bless Shehecheyanu on circumcision for it is more common; Pidyon ha'Ben is not so common.

Rosh (ibid. and Kidushin 1:41): The Ge'onim enacted a long Berachah (Asher Kidesh Ubar bi'M'ei Imo...) for the Kohen to make. In France and Ashkenaz we do not say it, for the Kohen does no Mitzvah, he merely receives Matanos Kehunah. Also, we may not enact new Berachos after the sealing of the Talmud. Also, the fetus is not Kodesh bi'M'ei Imo, rather, when it is born!

The Ramban (Hilchos Bechoros 62b) objects because the Berachah connotes that a Bechor for Pidyon must be a Bechor for inheritance, and this is not true.

Question: Why was a Berachah composed for a Kohen accepting Pidyon ha'Ben, but not for other Matanos Kehunah?

Answer (Rivash ibid.): Regarding other Matanos Kehunah, the Kohen merely receives; here, he acquires the son and returns him to the father.

Question (Taz OC 46:7 and Pri Chodosh 51:1): The Rosh himself brings a Berachah on Besulim enacted by Ge'onim after the Gemara! Baruch she'Amar is not in the Gemara, and the Rosh does not question it!

Answer (Taz ibid.): We do not abolish a custom to say a Berachah of the Ge'onim; perhaps they had a source from the Gemara. Regarding Pidyon ha'Ben the Rosh explains why certain places do not say it. This is why the Rema defends the custom to bless 'ha'Nosen la'Ya'ef Ko'ach', even though it is not in the Gemara.

Gra (46:4 DH Ach): Custom overrides Halachah.

Poskim

Shulchan Aruch (YD 305:10): When one gives the money to the Kohen he blesses Asher Kidshanu...Al Pidyon ha'Ben, and then Shehecheyanu, then he gives the money to the Kohen. One who redeems himself blesses Lifdos ha'Bechor and Shehecheyanu.

Rema: Some say that even one who redeems himself blesses Al Pidyon ha'Ben; this is our custom.

Source #1 (Ran 7b DH Avi): We bless 'Al' because it can be done by a Shali'ach.

Source #2 (Rivash ibid., brought in Taz 8): We do not bless with a 'Lamed' because another person (the Kohen) is needed for the Mitzvah.

Chachmas Adam (150:7): One who redeems himself blesses 'Al Pidyon ha'Ben'.

Kitzur Shulchan Aruch (165:8): One who redeems himself blesses 'Al Pidyon Bechor' and Shehecheyanu.