Sanhedrin 21a - Question: "Va'Yikach David Od Pilagshim v'Nashim" - what is the difference between Pilagshim and Nashim (wives)?
Answer (Rav Yehudah): Wives have a Kesuvah and Kidushin, Pilagshim do not have a Kesuvah or Kidushin.
Rambam (Hilchos Melachim 4:4): A king may (forcibly) take wives and Pilagshim from anywhere in Yisrael. Pilagshim have no Kesuvah or Kidushin. He acquires her through mere seclusion, and she is permitted to him. A commoner's only Heter for a Pilegesh is an Amah Ivriyah after Yi'ud (when her master says that he is marrying her h0m or to his son).
Question (Keneses ha'Gedolah): Yi'ud makes Kidushin!
Aderes Eliyahu (Reish Mishpatim): This shows that a Pilegesh has Kidushin.
Answer #1 (Chacham Tzvi, Tosfos Chodoshim 24): After Yi'ud, she is a full wife - "She'erah Kesusah v'Onasah Lo Yigra"! The Rambam means that this is the only seeming Heter for Bi'ah without Kidushin. Really, Yi'ud is Kidushin.
Answer #2 (Aruch ha'Shulchan EH 26:8): One Tana in the Mechilta explains that if a son did Yi'ud, he will treat her like one whom a master was Meya'ed, i.e. a Pilegesh. "If he will take another, he will not diminish his wife's She'er, Kesus v'Onah". The Rambam teaches that Yi'ud is not regular Kidushin!
Answer #3 (Merumei Sadeh Sanhedrin 21a DH Pilagshim): Pilagshim are without Kidushin, i.e. Kidushin is not needed before seclusion and Bi'ah.
Answer #4 (Devar Avraham, Hashmatah after 3:31): The Rambam means that an Amah after a Yi'ud is like a Pilegesh in one way: she has no Kesuvah.
Rebuttal (of Rambam - Rashba Teshuvah 284 attributed to Ramban): One designates a Pilegesh for himself. Surely this is permitted! We forbid Zenus in Yisrael lest it lead to incest (if no one will know who is a child's father). David married Pilagshim. We have no source to distinguish commoners from kings. Kalev and Gid'on had Pilagshim. We cannot say that a judge over the Tzibur is like a king, for then the law is not uniform. If a Pilegesh were forbidden, why did the Ba'al of Pilegesh b'Giv'ah openly seek her, and rebuke those who sinned with her? We have no source that Chachamim forbade it. We forbid a Kalah without Berachos (of Nisu'in) like a Nidah, i.e. if one wants to marry her and forbid her to all others. If her Kesuvah is less than 200, she thinks that he seeks to divorce her. The Rambam does not forbid her to a commoner. (The Ramban's text of the Rambam was lacking - Kesef Mishneh Ishus 1:4.)
Defense (Gra EH 26:7): The Rambam permits Amah Ivriyah after Yi'ud. This explains Kalev, Gid'on,... Rashi's text says that a Pilegesh has Kidushin. In the Yerushalmi, Tana'im argue about whether a Pilegesh has a Kesuvah. It seems that all agree that she has Kidushin. The Medrash calls Avshalom's Bi'ah with David's Pilagshim 'Giluy Arayos', and it forbade them to David. Avishag was permitted to Shlomo (Sanhedrin 22a). We did not say so about David's Pilagshim. The father of Pilegesh b'Giv'ah was called the Ba'al's father-in-law. We understand the Ba'al's wrath if there was Kidushin. How can the only difference between a Pilegesh and a wife be a Kesuvah, which is mid'Rabanan? Rashi could hold that Kesuvah is mid'Oraisa, like R. Shimon ben Gamliel. Alternatively, 'Kesuvah' refers to stipulations of the Kesuvah.
Radvaz (also Teshuvah 4:225): If there were Kidushin, Yisrael would not have allowed Avshalom to do so! Pilegesh b'Givah is called a wife, because men treated Pilagshim like wives. At the time, it was permitted. If there were Kidushin, her Zenus would have forbidden her to him! Later, Chachamim forbade Pilagshim to commoners, lest she be ashamed to immerse.
Note: Gitin 6b says that she was not really Mezanah. Tosfos (DH Zevuv) says that the practice was not to take back a Pilegesh who was Mezanah.
Rivash (398): After David's Beis Din decreed against seclusion with a Penuyah, we do not find Pilagshim in Tanach except for David's (who were previously permitted) and Shlomo (who transgressed even marrying too many wives). David's were called 'living widows', i.e. there was Kidushin.
Taz (26:2): If there is Kidushin, it is Bi'as Zenus because she does not have a Kesuvah of 200! Surely Avraham did not transgress!
She'alas Ya'avetz (2:15): When they were away, some Amora'im would get a woman 'for a day' (Yoma 18b), i.e. a Pilegesh. There is no Kidushin. She is designated for him, and hence forbidden to others due to Kedeshah, and (also for three months after leaving him), to avoid doubts about fatherhood.
Rambam (Hilchos Ishus 1:4): Anyone who has Bi'ah with a woman for Zenus without Kidushin is lashed mid'Oraisa for "Lo Tihyeh Kedeshah".
Rebuttal (Ra'avad): If she designates herself for one man, there is not even a Lav. This is the Pilegesh of the Torah. Some forbid, and lash for Kedeshah. Some texts in Sanhedrin say 'Pilagshim have Kidushin, but not a Kesuvah.'
Rivash (398) and Magid Mishneh: The Rambam had this text. Or, even our text could mean 'they do not have Kidushin and Kesuvah', just Kidushin.
Rebuttal (Kesef Mishneh): The Rambam in Hilchos Melachim explicitly says that a Pilegesh is without Kidushin.
Note: See Answer #2 above (she does not have regular Kidushin).
Rosh (Kesuvos 1:12): If one fulfilled Peru u'Rvu through a Pilegesh, he is not obligated to be Mekadesh a woman.
Shulchan Aruch (EH 26:1): Bi'as Zenus not l'Shem Kidushin, or even Bi'ah l'Shem Ishus alone (without witnesses), does not make her his wife. This is even if he designated her for himself. To the contrary, we force him to expel her!
Gra (3): When he designated her, it is known that they have Bi'ah. This is like witnesses. However, the Bi'ah was not l'Shem Kidushin.
Rema: Surely she is ashamed to immerse. They have Bi'ah while she is Nidah.
Source (Teshuvas Rosh 32:13, in Beis Yosef Sof Siman 22): If a Penuyah serves in Reuven's house and there are unceasing rumors that they are secluded together, her family may protest. It is a blemish to them that she is his Pilegesh. Beis Din forces him to expel her, she is ashamed to immerse.
Bach (DH u'Mah): The Tur says that the Rosh forbids Pilegesh. Her family can protest because the Isur is a blemish to them! The Rosh's text holds that a Pilegesh has Kidushin. Without Kidushin, it is forbidden.
Rebuttal (Beis Shmuel 2 and Chelkas Mechokek 2): The Rosh and Tur do not forbid Pilegesh. It is a blemish, for it is shameful. If the Lav of Kedeshah applied, Beis Din would separate them even without concern for Nidah! The Rosh in Kesuvos proves that he permits Pilegesh.
Rebuttal (Korban Nesan'el 5): Perhaps the Rosh in Kesuvos means that if he had children through a Pilegesh (b'Isur), he need not be Mekadesh a woman!
Chelkas Mechokek (2): It seems that even the Rambam does not lash for a Pilegesh designated to one man. 'One who has Bi'ah with a woman for Zenus without Kidushin is lashed' connotes that if she was designated for him, even without Kidushin, at least there is no Lav. In Hilchos Melachim (4:4) he forbids a Pilegesh to a commoner. He did not mention lashes.
Rema: If he designates a woman and she immerses for him, some permit. This is the Pilegesh of the Torah. Some forbid, and lash for "Lo Tihyeh Kedeshah."
Gra (6): The first opinion is like our text in Sanhedrin (that a Pilegesh has no Kidushin or Kesuvah), and that of the Rambam and Ramban.