Is a person (A) allowed to contact someone (B) who A knows is currently employed, and ask if B is open to hearing about a new work opportunity? A could be a recruiter or an employee at another company who is interested in hiring B.

ב"ה Shalom The issue is discussed extensively, by the renowned Rabbi Dovid Lichtenstein who deals with many contemporary Halachic issues in his books and other forums of "Headlines". See the link below for full details https://staff.ncsy.org/education/education/material/wgFT2jzYQg/poaching-employees-from-rival-companies/ To sum up, Rabbi Lichtenstein quotes Rav Tzvi Shpitz: "Rav Tzvi Shpitz, in his Mishpetei Ha’Torah (1:50) rules that poaching is forbidden if this would result in significant damage to the competitor’s livelihood. Rav Shpitz adds, however, that although one may not directly work to recruit employees under such circumstances, it is permissible to advertise employment opportunities in the hope of arousing the interest of the competitor’s current staff." Rabbi Lichtenstein in his own conclusion says: Practically speaking, an employee may always quit his job. There is a machloket between Rashi and Tosfot if one may poach an employee, based off of their opinions regarding the case of the a "poor man in the trash heap."(See Kiddushin 59a) Rashi says it is forbidden, and Tosfot allows it. Many poskim hold like Tosfot but the Aruch Hashulchan (Choshen Mishpat 237:5) says the Shulchan Aruch ( Choshen Mishpat 237:1) holds like Rashi. Tosfot themselves seem to say poaching is forbidden when the competitor’s livelihood is threatened. Addendum: Tammuz 5782 After having written this answer, vol. 50 of the Responsa Yashiv Yitzchak by Rav Yitzchak Shechter Shlit"a from Kiryat Zans, Netanya was published in regard to a similar question of persuading a caregiver already working for one person to work for another. Rav Shecter's answer was similar saying it was forbidden to poach the employee, basing himself on the Rema (חו"מ סי' רל"ז) and other major Poskim. ישיב יצחק חלק נ' סימנים מג-מ"ה ,מנחת פיתים סי' י"ד, על חושן משפט סי' רל"ז:א, מנחת יצחק חלק ו סי' ע"ז All the best