Min haTorah the prohibition against carrying is from a reshus hayachid [private domain] to a reshus harabbim [public domain] and vice versa or the moving of an object four amos in a reshus harabbim.
Chazal added a prohibition against carrying in a domain known as a karmelis [an area that cannot be classified as a reshus hayachid, since it does not have the required mechitzos or as a reshus harabbim, because it does not meet the necessary criteria]. Since there are similarities between a reshus harabbim and a karmelis, Chazal prohibited carrying between any two domains as well as within any domain other then a reshus hayachid itself  in order to prevent any inadvertent transgression of the laws of carrying in a reshus harabbim (Shabbos, 6a see Rashi ad loc. and Shulchan Aruch, Orach Chaim 346:1).
Since the only domain in which carrying remains permissible is a reshus hayachid, our primary concern when planning the construction of an eruv is that we be able to classify the area under consideration as a reshus hayachid.
 Me’d’rabbanan, even after determining that a halachically enclosed area is a reshus hayachid, the ability to carry therein is contingent on the residents forming a unified entity or eruvei chatzeiros. Since this requirement is me’d’rabbanan, Chazal were lenient and only necessitated a symbolic unified ownership. Depending on who joins this symbolic partnership, one or two methods must be employed: eruvei chatzeiros or sechiras reshus.
Eruvei chatzeiros, the merging of two or more courtyards or apartments ― used when dealing with Shomer Shabbos residents ― works in the following manner: Pieces of bread or matzoh that were contributed by all those residing within the eruv are placed in one of the enclosed houses, or alternatively one resident may take his own matzoh and grant the other residents ownership of it [zechiah]. These methods enable us to symbolically view all the residents who reside within this eruv as having united to dwell in the house in which the matzoh is kept (Tur and Shulchan Aruch, O.C. 366:1, 7).
Sechiras reshus, the rental of a domain ― usually used when dealing with neighbors who are not Shomrei Shabbos, either because they are not observant or not Jewish. Through the symbolic rental from the owners of dwellings in an enclosed area, one can acquire the right to carry therein. So too, in public areas ― such as when enclosing a city that has numerous private residences and public streets ― one can acquire the right to carry through the rental of these privileges from city officials (ibid., 382:1, 391:1).
Related to the forming of eruvei chatzeiros is the issue of being mocheh. Some claim that when a frum yid is mocheh it disrupts the unity needed for eruvei chatzeiros. However most poskim maintain that this does not create a problem (U’Bacharta B’Chaim, siman 123; Taanugei Yisroel, 2:42:7; Tzitz Eliezer, 19:17; Emek HaTeshuvah, 4:21-23, and Kovetz Ohr Yisroel, vol. 18, 21). Additionally, since we contract sechiras reshus from the city and state governments, even those poskim who have a problem with sechiras reshus when a frum yid is mocheh would allow carrying based on the power of eminent domain (Kovetz Ohr Yisroel vol. 32-33; and read carefully Divrei Yatziv 2:173:6).
Besides for which, all are in agreement that regarding eruvei chatzeiros we maintain halachah k’divrei hameikil (Eruvin, 46a, 80a). Therefore, there is no reason to acknowledge someone who is mocheh since there are many reasons to be lenient. [Actually, there are many poskim who posited that we pasken halachah k’divrei hameikil even in issues regarding mechitzos (see Part 1: Halachah K’Divrei Hameikil B'Eruvin).]
To include sechiras reshus among Rav Moshe Feinstein zt”l’s issues with reshus harabbim is disingenuous because the Klausenberg Rebbe zt”l (Divrei Yatziv, 2:173:5), who is the one who took issue with sechiras reshus ― when a frum yid is mocheh ― stated clearly that today there is no reshus harabbim. So of course his position on the issue of reshus harabbim would be that there is no reason to be mocheh. These detractors of eruvin, however, claim that they are mocheh because according to the way they understand Rav Moshe there is a reshus harabbim today; but Rav Moshe never had a problem with sechiras reshus. When the Manhattan eruv was established a frum yid was mocheh (Divrei Menachem, O.C. vol. 2, p. 95), and neither Rav Moshe nor anyone else had a problem with sechiras reshus in that situation. For that matter Rav Moshe never had a problem with sechiras reshus when a frum yid is mocheh. Rabbi Shlomo Pearl shlita stated (in a shiur he gave in Agudath Israel Bais Binyomin, Nov. 14, 1999 tape #325) that he had spoken with Rav Dovid Feinstein shlita about being mocheh against the eruv [in Boro Park]. Rav Dovid made it clear that no one should be mocheh. Additionally, Rav Dovid was unequivocal in his declaration that even if a gadol, in order to negate the eruvei chatzeiros, was to declare that he is mocheh against the eruv, he “must be crazy.”