Gemara

Contradiction: It says "Ashrei Nesuy Pesha Kesuy Chato'oh"; it also says "Mechaseh Fesha'av Lo Yatzli'ach"!

Resolution #1 (Rav): One should openly confess a known transgression, but not one that was not known to others.

Resolution #2 (Rav Nachman): One should openly confess a transgression against people, but not Bein Adam la'Makom.

Beraisa - R. Yehudah ben Bava: One must detail his sins when confessing. This is why Moshe said "Ana Choto ha'Am...va'Ya'asu Lahem Elohei Zahav";

R. Akiva says, "Ashrei...Kesuy Chato'oh" (it is better not to detail his sins). The reason Moshe mentioned Yisrael's sin was like R. Yanai taught:

R. Yanai: "V'Di Zahav" - Moshe said, the gold and silver that You gave to Yisrael (when they left Mitzrayim) until they said 'Dai' (enough!) caused them to make the golden calf.

Sotah 7b - Question: Granted, Yehudah (publicly) confessed so that Tamar should not be burned. Why did Reuven (publicly) confess?

Rav Kahana taught that one who publicizes his sins is brazen -- "Ashrei...Kesuy Chato'oh"!

Answer: He did so so that his brothers would not be suspected.

Sotah 32b - R. Yochanan, citing R. Shimon bar Yochai): It was instituted to say Tefilah quietly, to avoid embarrassing (penitent) sinners (who confess their sins).

Rishonim

Rif and Rosh: One should openly confess a transgression against people, but not one that was only against Hash-m.

Ran: One must openly confess a transgression against people in order to get others to persuade the victim to forgive him.

The Rif and Rosh bring the argument of R. Yehudah ben Bava and R. Akiva about whether or not one must detail his sins.

Rambam (Hilchos Teshuvah 2:5): It is praiseworthy for a penitent to confess in public and reveal his sins to them. He should say what he sinned against others, and that he repents and regrets. If one covers his sins his Teshuvah is incomplete; "Mechaseh Fesha'av Lo Yatzli'ach". This refers to sins against people, but one need not publicize a sin Bein Adam la'Makom. It is audacious to reveal it. Rather, he should repent to Hash-m, detail his sins in front of Him and confess in front of Rabim without specifying (the sins). It is good for him that they are not known - "Ashrei...Kesuy Chato'oh".

Ra'avad: Likewise, he should openly confess known transgressions, even Bein Adam la'Makom. Just like the sin was known, he must publicize the Teshuvah and be ashamed in public.

Support (Migdal Oz): Achan was told to publicly confess (Yehoshua 7:19). Shimon ben Shetach publicly confessed (Makos 5b). If an Ir Miklat wants to honor a murderer, he must tell them that he killed (Makos 12b).

Kesef Mishneh: Also the Rambam says not to reveal only sins Bein Adam la'Makom that are not known. He says that it is good for him that they are not known; this shows that he discusses sins that were not known! There is a difference between sins Bein Adam l'Chavero and Bein Adam la'Makom. The former one must publicize to obtain pardon from the hurt party. But one need not publicize sins Bein Adam la'Makom. Perhaps someone will hear who did not know until now, and there will be Chilul Hash-m. However, if charged he may not deny them.

Note: It appears that Migdal Oz and the Kesef Mishneh consider a sin to be known whenever people knew about it, even if they did not know who did it. Viduy causes a Chilul Hash-m by publicizing that someone was brazen enough to transgress Hash-m's will. (Nevertheless Hash-m permits it to enable people to do a proper Teshuvah.) If the sin is already known, the Kidush Hash-m through in public Teshuvah overweighs the small Chilul Hash-m of mention of the sin. The Gemara asked why Reuven confessed because only Yakov knew about the sin (see Rashi Sotah 7b DH d'Lo).

Lechem Mishneh: The Rambam holds that the two answers disagree, and he rules like the latter. One must openly confess only sins Bein Adam l'Chavero. The Ra'avad holds that both answers are true, so also known transgressions must be openly confessed, even if they are Bein Adam la'Makom.

Rambam (ibid. 3): In Viduy one must detail his sins - "Ana Choto ha'Am..."

Source - Kesef Mishneh: Normally, the Halachah follows R. Akiva against any colleague. However, Rav obligates openly confessing a known transgression. This is unlike R. Akiva, for Chet ha'Egel was known, nevertheless he needed to find another reason why Moshe mentioned it. Even though Rav Nachman could hold like R. Akiva, and the Rambam rules like Rav Nachman's answer to Rav's question, we learn from Rav that the Halachah does not follow R. Akiva. However, the Rif and Rosh disagree. A support for the Rambam is that there is an argument about whether or not one should confess sins from which he already repented and confessed on a previous Yom Kipur. Both opinions agree that one must detail his sins.

Rebuttal (Lechem Mishneh): Perhaps the argument about confessing sins that one already confessed refers to sins Bein Adam l'Chavero; all agree that one must detail them. Others rule like R. Akiva because the Halachah follows R. Akiva against a colleague. The Kesef Mishneh's other reason to rule like R. Yehudah ben Bava is flimsy. Nevertheless, one may rely on it, because the Yerushalmi says that R. Akiva only exempts from detailing sins. He does not forbid it.

Question: The Rambam brings "Ana Choto ha'Am..." to prove that one must detail his sins. The Egel was Bein Adam la'Makom. One need not reveal such sins!

Answer (Lechem Mishneh): Even R. Yehudah ben Bava does not require revealing sins to people. The Chiyuv to detail them is when confessing to Hash-m.

Sha'arei Teshuvah (1:18): Even though a Ba'al Teshuvah should not reveal his sins, he must confess them - "Chatasi Odi'acha va'Avoni Lo Chisisi". Sometimes "Mechaseh Fesha'av Lo Yatzli'ach". For sins Bein Adam l'Chavero one must ask forgiveness. If a sin Bein Adam la'Makom became known this is Chilul Hash-m; one must lament such sins publicly to make a Kidush Hash-m,

Poskim

Shulchan Aruch (OC 607:2): One need not detail his transgressions when confessing, but it is permitted. If he confesses quietly it is proper to detail them (this is how Magen Avraham (2) explains the Shulchan Aruch).

Gra (DH v'Im Misvadeh): The Shulchan Aruch forbids detailing when confessing out loud, like the Rema below. The Shulchan Aruch means, one need not detail his transgressions. It is permitted if he confesses quietly, and then it is proper to detail them.

Beis Yosef (DH v'Af and DH ul'Inyan): The Rif and Rosh bring Rav Nachman, but not Rav. Presumably, they hold that the Halachah follows R. Akiva, therefore the Halachah does not follow Rav. Presumably, even R. Akiva does not forbid detailing sins. He just holds that it is better not to. Perhaps he agrees that it is proper to detail sins quietly, for he feels more shame, yet this is still "Ashrei Nesuy Pesha Kesuy Chato'oh".

Rebuttal (Magen Avraham 2): The Amora'im do not argue with the Tana'im. Rather, the Tana'im argue about detailing sins silently to Hash-m. Therefore, they did not distinguish whether or not the sin is known. The Amora'im argue about detailing out loud sins that are known (or Bein Adam l'Chavero). Rashi (Chulin 41b DH Lo Shanu) says that people would publicize a Shogeg sin to increase their shame and get a better Kaparah. This is difficult. Chachamim enacted to pray silently to avoid shaming sinners, just like the Torah commands to slaughter Chatas in the same place as Olah to spare sinners from shame (Sotah 32b). Also, "Ashrei Nesuy Pesha Kesuy Chato'oh"!

Question: The Gemara challenged R. Akiva from "Ana Choto ha'Am..." This shows that R. Akiva forbids detailing sins!

Answer (Taz 2): The Beis Yosef says that R. Akiva holds that it is better not to detail sins. The Gemara challenged this, for surely Moshe did what is best. It seems that the Shulchan Aruch makes a compromise ruling. Since R. Yehudah ben Bava says that one who does not detail transgresses, and R. Akiva does not forbid, all permit to detail sins.

Taz (1): All agree that one must detail his own sins. The argument is whether or not one must detail sins when confessing b'Tzibur on behalf of the Tzibur.

Yad Efrayim (DH kedi'Chsiv): If one can prevent a loss to others by detailing his sin out loud, he should do so, like we find regarding Reuven (Sotah 7b).

Rema: An individual praying out loud or a Shali'ach Tzibur should not detail sins.

Beis Yosef (DH v'Nir'eh Li): The Shali'ach Tzibur may not detail sins, for this is like accusing the Tzibur of particular sins.

Mishnah Berurah (9): This applies even to sins Bein Adam l'Chavero.

Rema: The Viduy we say according to Aleph-Beis is not considered detailing sins because everyone says the same thing. It is like prayer.