Avodah Zara 2:1

It is prohibited to lodge an animal in an idolator’s inn because they are suspected of illicit acts; a woman may not be alone with them for similar reasons. A man may likewise not be alone with them because they are also suspected of bloodshed. A Jewish woman may not deliver an idolatrous woman’s baby because she would be delivering it for idolatry but an idolatrous woman may deliver a Jewish woman’s baby. Similarly, a Jewish woman may not nurse an idolatrous woman’s child but an idolatrous woman may nurse a Jewish woman’s child in the Jewish woman’s home.

Avodah Zara 2:2

Idolators may be hired to heal one’s animals but not to treat humans, nor may one allow them to cut one’s hair (out of a concern that they might slit the Jew’s throat), not even in public; this is the opinion of Rabbi Meir. The Sages permit it in public but not in private.