Gemara

20b (Mishnah): A millstone's lower stone must be distanced three Tefachim. The upper stone will be four Tefachim away.

This is due to the noise.

(Mishnah): If one set up a store in a Chatzer, others can stop him because the noise of people entering and leaving disturbs their sleep. Rather, he may make Kelim in the Chatzer and sell them in the market.

If their sleep is disturbed by the noise of his hammer, millstone or children (coming to the store) they cannot complain.

Question: What is the difference between the clauses?

Answer #1 (Abaye): The Seifa teaches that people of another Chatzer cannot complain.

Objection (and Answer #2 - Rava): If so, it should say so explicitly! Rather, the Seifa (does not discuss a store, rather, it) discusses one who teaches Torah to children, after the enactment of Yehoshua ben Gamla.

23a: Bloodletters used to sit under Rav Yosef's date trees. The Kelim attracted ravens, which damaged the dates. Rav Yosef said that the bloodletters must leave, for Rav Nachman taught that there is no Chazakah for damage.

Abaye: Rav Mari taught that that refers to smoke, and Rav Zevid taught that it refers to a Beis ha'Kisei (a privy. It does not apply to other things!)

Rav Yosef: I am very sensitive. The ravens bother me like smoke and a Beis ha'Kisei.

57b (R. Yochanan): Partners can object against each other regarding any use of the Chatzer, except for laundry. This is an enactment lest women need to launder in the river.

Rishonim

The Rif and Rosh (2:6) bring the Gemara verbatim.

Nimukei Yosef (DH mi'Kol): The Mishnah says 'the noise of people entering.' Really, it is not due to noise, rather, to increased traffic. The Yerushalmi clearly says so.

Nimukei Yosef (DH Gemara Omar): Abaye taught that people of another Chatzer cannot complain. This shows that one in another Chatzer of the Mavuy cannot complain about increased traffic. Rava refuted Abaye's Perush, but he did not argue with this law (that one cannot complain about increased traffic in a Mavuy). Above, we said that people in a Mavuy can prevent someone from making an opening to the Mavuy. We must distinguish between increased traffic of people coming to buy from increased traffic from another Mavuy.

Nimukei Yosef (DH l'Tinokos): Even though he could teach the children in a Beis ha'Keneses or Beis Medrash special for this, we do not require him to do so, to strenghen the enactment so that teachers will be available.

Hagahos Ashri: In every case, one may protest against one who wants to establish a profession. Rav Nachman taught that there is no Chazakah for damage, but we established this to refer to smoke or a Beis ha'Kisei. One can get a Chazakah for other damages.

Rambam (Hilchos Shechenim 6:12): If one set up a store in a Chatzer, the neighbors can stop him because the noise of people entering and leaving disturbs their sleep. Rather, he may make Kelim in his shop and sell them in the market. They cannot protest that their sleep is disturbed by the noise of his hammer or millstone, for he made a Chazakah. Similarly, he may teach Torah to Yisrael children, and the partner cannot complain that he cannot sleep due to the children.

Magid Mishneh: The Rambam explains that this is when he already made a Chazakah, but at the beginning they can stop him. The Rambam was bothered why they cannot protest. A hammer disturbs sleep more than people entering and leaving! Also, the Mishnah says 'a store in the Chatzer', which connotes that it is already there. It does not say 'one who seeks to make a store' or 'one may not make a store.' Rather, it discusses one who already has a store. It teaches that his Chazakah does not help for others, i.e. people entering and leaving, like he said in the name of Ge'onim in Halachah 11:5. His Chazakah helps for himself to do his own work. The Rashba disagrees and says that others cannot protest about the noise of the people entering and leaving, for they cannot protest about the noise of his hammer or millstone! Rather, they can protest that the increased traffic in the Chatzer increases the distance they must walk. He says that even if they permitted this, they can say 'we thought that we could tolerate it, but now we cannot.' He derives this from the words 'a store in a Chatzer', like I said above.

Poskim

Shulchan Aruch (CM 156:2): If one set up a store in a Chatzer, the neighbors can stop him because the noise of people entering and leaving disturbs their sleep. Rather, he may make Kelim in his shop and sell them in the market. They cannot protest that their sleep is disturbed by the noise of his hammer or millstone, for he made a Chazakah and they did not protest.

SMA (9): This is unlike grinding grits (157:14), which others can protest about, for it makes a great noise and shakes walls, which is a great damage.

SMA (10): They can protest against the other matters, which are in the Chatzer, but not against a hammer or millstone, which are in his house. The Shulchan Aruch says in Sa'if 4 that Chazakah helps for Melachah, but not for people entering and leaving.

SMA (11): The Rambam says that Chazakah helps. Three years are not needed. It suffices that it is recognized that he did Melachah, they heard and did not protest.

Gra (9): The Tosefta (1:3) says that if the store was there first, others cannot protest.

Rema: Some say that whatever he does in his house even l'Chatchilah they cannot protest.

Beis Yosef (DH v'Amai): R. Yo'el (in Mordechai 512) says that they cannot protest against his hammer in his house, but in the Chatzer, partners can protest about anything except for laundry, like it says in 57b. R. Yerucham and the Semag say that before he gets a Chazakah, neighbors can protest against a hammer in his house. It seems that the Rambam agrees. The Ramban holds like the Rashba (they can protest only due to increased traffic), and it seems that also Tosfos does.

SMA (12): This is unlike the Mechaber, who permits only if he already made a Chazakah.

Rema (ibid.): This refers to healthy people. Sick people can protest if the noise harms them.

Gra (11): We learn from 23a.

Shulchan Aruch (3): Similarly, he may teach Torah to Yisrael children, and the neighbors cannot complain that they cannot sleep due to the children learning from their Rebbi. Similarly, they cannot protest about any Mitzvah.

Beis Yosef (DH u'Mah she'Chosav Rabeinu): The Ritva (Sof Teshuvah 27) says that likewise they cannot protest if he sets up a Medrash to spread Torah or expound to the Rabim to glorify Torah, lest it be forgotten.

SMA (13): This connotes that there is no limit to the number of children. The Tur says so explicitly. This is unlike the Ramban, who says that Rashi holds that it is forbidden if there are more than 50.

Gra (12): This is l'Chatchilah (neighbors cannot protest).

Pischei Teshuvah (9): The Levush says that we discuss people of the Chatzer or Mavuy (alley) protesting, but one who lives in the same room or Beis ha'Choref (a room with little ventialltion, for the winter) can prevent another even from teaching Torah to Yisrael children, for most people cannot bear this. He did not become partners in the room with this intent, unless he explicitly stipulated.