Before us is Mishnah 6 in the third chapter of Maseches Chagigah, which deals with the credibility of amei ha'aretz in matters of purity: the credibility of the tax collectors and of thieves returning what they stole, and the credibility of amei ha'aretz in Jerusalem.
"Hagaba'in shenichnesu l'soch habayis":
The gaba'in were Jews who collected taxes on behalf of the government, and since they were amei ha'aretz, anything they touched was liable to be considered tamei. These collectors would enter people's private homes, and the purpose of their coming was to take collateral for the taxes that were owed.
Alongside them the Mishnah lists another group: "v'chein haganavim shehecheziru es hakeilim" - thieves who stole vessels and afterward repented and came to return those vessels to their owners.
Regarding both of these groups the Mishnah rules: "ne'emanin lomar lo naganu" - the collectors did indeed touch the vessels they took as collateral, but with regard to any other vessel that they did not take, they are believed to say that they did not touch it, and those vessels remain in their presumed state of purity.
As for the thieves who return the vessels, they certainly touched the vessels themselves. However, when the case involves earthenware vessels, which do not become tamei from their outer surface, they are believed to say that they did not touch the inside of the vessels but only the outside.
"U'virushalayim ne'emanin al hakodesh":
In Jerusalem itself, the amei ha'aretz were believed even with regard to consecrated food, and in fact especially with regard to it, since in matters of kodesh people were careful to ensure that it remained in its purity.
The Gemara brings a beraisa which explains that in Jerusalem the amei ha'aretz were believed regarding all earthenware vessels, both large and small, to say that they are pure for kodesh. In the previous Mishnah we learned that from the city of Modi'im inward the amei ha'aretz were believed regarding earthenware vessels, but that was specifically with small earthenware vessels. Our Mishnah deals with Jerusalem itself, which is also located within the area from Modi'im inward, and there they were believed regarding all earthenware vessels.
The reason for this: earthenware vessels were not manufactured in Jerusalem, because people did not want the kilns fired there on account of the pollution they caused. As a result, earthenware vessels were not plentiful there, while they were needed for the sacrifices. Therefore the Sages were especially lenient and believed the amei ha'aretz within Jerusalem itself regarding the use of all earthenware vessels.
"U'vish'as haregel af al haterumah":
At the time of the festival - on Pesach, on Shavuos and on Sukkos - the amei ha'aretz in Jerusalem were believed even with regard to Terumah, and not only with regard to kodesh.
In summary: In this Mishnah we learned that the tax collectors and the thieves who return vessels are believed to say that they did not touch the remaining vessels, and with earthenware vessels they are believed to say that they touched only the outside; that in Jerusalem the amei ha'aretz are believed regarding kodesh and regarding all earthenware vessels, because earthenware vessels were not produced there; and that at the time of the festival their credibility extends even to Terumah.