Demai, Chapter 4, Mishnah 3. The entire concept of demai rests on a doubt: we do not know whether the am ha'aretz separated the tithes or not. Because of this doubt, the Levi has no monetary claim to the first tithe, since "the one who wishes to extract from his fellow must bring the proof" - it is up to the Levi to prove that the tithes were in fact not separated and that he is owed what is coming to him; and since he cannot prove this, he does not receive the tithe.
Why, then, does the chaver separate first tithe at all:
A Jew who is a chaver and buys produce from an am ha'aretz is required to separate first tithe, since this is the only way to remove the concern of tevel - whether or not this tithe is fit to be eaten by anyone. There are two reasons for this:
The concern of tevel: Without separating first tithe, the produce remains in the category of tevel, which is an object of prohibition by Torah law, and is forbidden to be eaten until all the tithes have been separated from it.
Terumah of the tithe: Terumah of the tithe, which has the status of Terumah and is forbidden to be eaten by anyone other than a kohen, can only be separated after the first tithe has been set aside. Therefore he separates the first tithe in order to reach the Terumah of the tithe, except that he is not obligated to give it to a Levi.
The question regarding tithe for the poor:
In a year of second tithe, a person must bring the tithe up to Jerusalem or redeem it. But what is the law regarding tithe for the poor, and why should it be different? Clearly there is no obligation to give it to a poor person, for the owner of the produce can say to him: "Prove to me that the tithes were not separated; the am ha'aretz maintains that he did separate them, and the burden of proof is on you." The question is whether there is any obligation to separate it at all, and why there should not be such an obligation. This is the subject of our mishnah.
The view of Rabbi Eliezer:
Rabbi Eliezer says "ein adam tzarich likros shem lema'aser ani shel demai" - there is no need to designate a portion of the produce as tithe for the poor. The reason: amei ha'aretz were not suspected of failing to separate tithe for the poor, since all that is at stake is the permissibility of eating it, and there is no prohibition in eating it at all. They might say: "True, it belongs to a poor person, but I am not so particular about giving it to a poor person" - meaning that they are not so careful about robbing the poor, and therefore they do separate tithe for the poor, they simply do not necessarily give it to a poor person.
The view of the Sages:
The Sages disagree: one must make a verbal designation and set aside the portion as tithe for the poor, but there is no obligation to actually separate it and give it to a poor person, because the poor person cannot prove that the tithe is truly coming to him.
A further explanation of the reasoning behind the dispute:
Some wish to explain the reasoning behind Rabbi Eliezer's view, that amei ha'aretz are not suspected of failing to separate tithe for the poor, in this way: the am ha'aretz will indeed separate it, since if he needs to, he can renounce ownership of all his possessions, and then he himself will be poor and will be able to receive the tithe for the poor. But the Sages hold that he would be afraid to actually carry out such a renunciation, lest someone grab his belongings and run off with them before he has a chance to reacquire them. Because of this concern, he would not view renunciation as a practical option, and therefore he would not separate tithe for the poor.
In summary: In this mishnah we learned that because of the doubt inherent in demai, neither the Levi nor the poor person can claim what is coming to him, and nevertheless the chaver separates first tithe - in order to remove the concern of tevel and in order to make it possible to separate Terumah of the tithe. Regarding tithe for the poor from demai the Tannaim disagreed: according to Rabbi Eliezer a person need not even make a verbal designation for it, and according to the Sages he makes the designation but need not actually separate it.