1:
Our mishnah contains two clauses. The first clause is concerned with the vicarious collection of
pe'ah: that is to say it is concerned with the possibility that one person might collect
pe'ah on behalf of another. Generally speaking, in the halakhic system it is possible for one person to take legal possession of something on behalf of another person who is absent. Usually, that is on the assumption that the other person has legal right to possession. In our mishnah the issue is different.
2:
The issue in our mishnah concerns the legal competence not of the eventual recipient but of the initial recipient. Legally, only someone who is indigent is entitled to take pe'ah from a field. (Of course, this also applies to the other sections of the 'poor law' such as gleaning and the Forgotten Sheaf: these are dealt with in later parts of this tractate.) If we assume that the initial recipient mentioned in our mishnah ("someone collects pe'ah") is himself (or herself) indigent then there is no reason why he or she cannot take legal possession of the pe'ah and there is no reason why he or she should not be able to do so on behalf of someone else who is indigent. But could this be the case if the initial recipient is not indigent?
3:
Rabbi Eli'ezer holds that a person of sound economic standing ('rich') is able to take possession of pe'ah on behalf of a named indigent person. The rest of the sages hold that such a person is not so entitled, that he is illegally in possession of the pe'ah and he must dispose of it by giving it to the very first person he meets who is legally entitled to it.
4:
The Gemara [pe'ah 18b] gives the reasoning of Rabbi Eli'ezer: in theory, our 'rich' person could be poor; all he (or she) needs to do is to legally abandon all their property and declare it to be ownerless and presto! they are poor and they are then entitled to the pe'ah. The rest of the sages see this as very spurious halakhic reasoning: can someone take possession of something that legally belongs only to a poor person on the assumption that they too could be poor if they wanted to be? Halakhah, of course, follows the reasoning of the sages and not that of Rabbi Eli'ezer. (I write 'of course' because this is the case in the overwhelming majority of instances of such disagreements. The view of the majority prevails over the view of the minority.)
5:
Another possibility that is raised in the Gemara is that the person who is the subject of our mishnah ("someone collects pe'ah") is the owner of the field himself (or herself) doing so on behalf of a named poor person. Here again, the litmus test applied is the financial state of the collector: if he is 'rich' he cannot take possession of the pe'ah; if he is poor he may do so and may subsequently pass it on to another poor person.
6:
The second clause of our mishnah is concerned with the status of pe'ah from the field which belongs to a non-Jew. It is the sanctity of the soil of Eretz-Israel that makes it subject to all the laws of the Torah. (This is why none of the laws that apply to things grown in the soil apply outside Eretz-Israel.) But non-Jews did acquire property in Eretz-Israel (and still do so). They themselves, obviously, are not subject to the laws of the Torah (because they are not Jewish). But does the land that they own thereby lose its sanctity? Or does the sanctity of the land revert to its original status when the non-Jew no longer possesses it?
7:
The view of the majority of the sages (which is reflected in the second clause of our mishnah) is that the sanctity of such land remains intact. If a Jew buys the land from a non-Jew all these agricultural laws apply once more. But as long as the land is held by the non-Jew these laws do not apply to his produce. But let us assume that such a non-Jew wishes to behave in the same manner as his Jewish neighbours and gives some of his produce to the indigent as if deriving from 'poor law'. We saw in Chapter 1 [Peah 012] that produce collected by the poor in this manner (pe'ah, gleanings etc) is not subject to tithing. Our present mishnah states that since the agricultural produce with which we are dealing here is owned by a non-Jew, when he donates it to a Jew it is subject to tithing because it is not true pe'ah etc. However, if the non-Jew declared it to be ownerless and just let the poor come to take it at their pleasure then it is free from the liability of tithing as is the case with any ownerless produce: the sanctity of the soil of Eretz-Israel re-asserts itself. (I have greatly simplified the halakhic situation here in order not to burden my explanation with matters that are not essential to an understanding of the mishnah itself.)
