Before us is a Mishnah in Maseches Yevamos dealing with testimony about the death of a husband - testimony that has the power to permit his wife to remarry.
The words of the Mishnah:
"Afilu shama min hanashim omros: meis ish peloni - dayo" - it is enough that a person heard women speaking among themselves and saying that a certain man died, in order to permit his wife to remarry. Even though they had no intention whatsoever of giving testimony, but spoke only in passing, that is sufficient.
"V'Rabbi Yehudah omer: afilu shama min hatinokos omrim - harei anu holchin lispod v'likbor ish peloni" - in the opinion of Rabbi Yehudah, even hearing children saying that they are going to, or coming back from, the eulogy and burial of a certain man is sufficient.
The Gemara clarifies that the children's words must include details about what took place at that funeral: how many rabbis were present, how many people participated, and that eulogies were delivered there. Without these details, there is reason to suspect that they gave the name of that man, whom they had heard about, to a small animal or an insect that died, and buried it. Therefore their words carry no weight unless they supplied details.
"Bein shehu miskavein u'vein she'eino miskavein" - whether the speakers intended to testify or did not intend to do so, one may rely on their casual conversation and permit the woman to remarry on its basis.
The view of Rabbi Yehudah ben Bava:
"B'Yisrael - ad sheyehei miskavein" - with a Jew, intent to testify is required.
"U'v'goy - im hayu miskavnin, ein eduso edus" - with a non-Jew the law is the reverse: if he intended to testify, his testimony does not count, and he is not fit to testify about such a matter.
The reason for this: a non-Jew is believed in this matter only when he did not intend to testify. If he said the words merely in passing, and not in order to permit the woman, he is believed. But if his intention was to permit her, and since he meant to affect the halachah, he is not believed. And when he was not seeking to affect the halachah but simply reported something in passing and incidentally, his words are taken seriously and we believe that the man indeed died, and his wife may be married off.
A different version of the Mishnah:
Some have a different reading in the Mishnah regarding the law of a Jew: not "ad sheyehei miskavein," that his intent is required, but rather "afilu miskavein" - even though he intended. This is in contrast to a non-Jew, who may not intend, and if he intended to say the words as testimony that would affect the halachah, his testimony is not believed.
According to this version, Rabbi Yehudah ben Bava comes primarily to teach us the law concerning a non-Jew, that he is not believed if he intended to affect the halachah. With a Jew, however, the law holds in both cases: he does not need intent, and even if he said the words in passing he is believed, and likewise if he intended to testify he is believed.
In summary: in this Mishnah we learned that even hearing women or children speaking about the death of a certain man is sufficient to permit his wife, provided that the children's words include the details of the funeral; that in the opinion of the first Tanna there is no difference between one who intends and one who does not intend; and that Rabbi Yehudah ben Bava distinguishes between a Jew and a non-Jew, with whom it is precisely the absence of intent that grants credibility - and according to another version, with a Jew he is believed in both cases alike.