Gemara

(Beraisa): If Reuven was working with Shimon's Taharos (Tahor food or Kelim) and he told Shimon that they became Tamei, he is believed. If he tells him 'your Taharos that I worked with on a previous day became Tamei', he is not believed.

(Abaye): He is believed only in the Reisha, for then it is b'Yado (he could make it Tamei) at the time he tells him.

(Rava): Reuven is believed only when he told Shimon the first time he saw him (after, according to Reuven, it became Tamei).

(R. Ami): If Levi told David 'the Peros that we were working with the other day became Tamei', letter of the law he is not believed.

(R. Asi citing R. Yochanan citing R. Yosi): The Torah believes him!

(R. Yitzchak bar Bisna): We learn from the inner Chata'os offered on Yom Kipur. No one else may be in the Ohel Mo'ed when the Kohen Gadol offers them. We could know that they became Pigul only if one witness (i.e. the Kohen Gadol) is believed! (We know that every Korban can become Pigul.)

A scribe said 'the Sefer Torah that I wrote for Ploni is Pasul because I wrote the occurrences of Hash-m's name Lo Lishmah (without intent).'

R. Ami: If Ploni has the Sefer Torah, you forfeit your wages through your admission, but you are not believed to disqualify the Sefer Torah.

A scribe said 'the Sefer Torah that I wrote for Ploni is Pasul because I did not tan the hide for the parchments Lishmah.'

R. Avahu: If Ploni has the Sefer Torah, since you are believed to forfeit your wages, you are also believed to disqualify the Sefer Torah.

In R. Ami's case, perhaps the scribe wrote it Lishmah, but hoped to disqualify the Sefer and still be paid for everything except for the names of Hash-m. Here, the scribe knows that he forfeits his entire wage. Presumably, he is telling the truth!

Kidushin 65b (Abaye): If Reuven told Shimon 'your Taharos became Tamei', or 'a man had Bi'ah with your ox (so it is Pasul for a Korban)' and Shimon was silent, Reuven is believed.

A Mishnah and Beraisa support these teachings.

Rishonim

The Rif and Rosh (5:13) bring the cases of the scribes.

Rambam (Hilchos Sanhedrin 16:6): Two witnesses are needed for lashes only at the time of the Aveirah. The Isur is established through one witness. E.g. if a witness said 'this is Chelev of the kidneys' or 'these Peros are Kil'ai ha'Kerem' or 'this woman is a divorcee or Zonah.' Later, if one ate them or (a Kohen) had Bi'ah with her in front of witnesses, he is lashed. However, if he contradicted the witness when he established the Isur, he is not lashed unless two witnesses establish the Isur.

Rosh (Gitin 5:8): There are 14 distinctions about when one witness is believed about Isurim. 1. If there was no Chazakah, one witness is believed to forbid something, e.g. to say that a piece of fat is Chelev (Yevamos 88a). 2. If there was a Chezkas Heter, one witness is not believed to forbid, e.g. that Taharos became Tamei (Gitin 54b). 11. One is believed about anything b'Yado, even if it had Chezkas Heter and the witness says that it is Tamei or Asur, and the owner contradicts him or says that he does not know. This is because anything b'Yado is like his own. It is not because he could Metamei it, for then he would have to pay. A scribe is believed to say that he wrote Hash-m's names Lo Lishmah as long as he has the Sefer Torah, even though if they were written Lishmah he cannot undo this. If one can tithe his produce to exempt another's, even though only a fool would do so, he is believed to say that another's Tevel was fixed. 12. If something was never b'Yado and it had Chezkas Heter, one cannot forbid if the owner contradicts him or says that he does not know. 14. Even if something is not now b'Yado, if the witness tells the owner at his first opportunity, he is believed.

Rosh (Mo'ed Katan 3:32): The Ramban says that we mourn based on one witness. Rabbeinu Meir objected. We concluded that one witness is believed only to permit. He can forbid only when the owner should know and he is silent, for this is like an admission.

Poskim

Shulchan Aruch (YD 127:1): If Reuven told Shimon 'your wine was poured to idolatry' and it was b'Yado to do so, he is believed. This is even after Reuven returned it and it is not b'Yado, if he told Shimon at the first opportunity, even if Shimon contradicts him. If it is not b'Yado and he did not tell Shimon at the first opportunity, he is not believed if Shimon contradicts him (Toch Kedai Dibur) or says that he does not know. If Shimon was silent, this is like an admission.

Rema: We counsel Shimon l'Chatchilah to say that he does not believe Reuven. Then, all agree that it is permitted, for an owner is believed about his property.

Beis Yosef (DH u'Mah she'Chosav Rabeinu she'Melamdim): R. Tam says that one need not be stringent even if he believes the witness like two. Rabbeinu Meir says that a Ba'al Nefesh should be stringent when the witness is trustworthy and we know that he does not seek to make him lose.

Shulchan Aruch (3): One witness is believed to permit Isurim, but not to be stringent.

Gra (21): This is difficult. If there is no Chazakah, he is believed in either case. If there is Chezkas Isur or Heter, he is not believed!

Shach (23): This means that one witness is believed to permit even without silence of the owner, which is like admission, but he cannot forbid without such admission. If there is no prior Chazakah, in any case we are stringent due to Safek. If it was Muchzak to be permitted, he is not believed to forbid.

Beis Yosef (DH Motzasi): I found written that R. Tam says that an owner who contradicts one witness is permitted, but others may not rely on the owner against the witness. One witness is not believed like two to forbid against Chezkas Heter is he is contradicted. The case of a Mikveh opposes this. One witness is believed to disqualify it (Kidushin 66b)!

Rema (ibid.): One witness is believed about anything without a Chazakah for Isur or Heter, even to forbid it.

Shach (25,26): He is believed even without the owner's admission, e.g. the owner is away or says that he does not know. Even though without the witness we would be stringent due to Safek, he is believed to make it Vadai Isur, and it cannot join another Safek to make a Sefek-Sefeka (two doubts)