In the US, Private cars were usually referred to as Business cars. These were indeed private. You only rode in them as a guest of the car's owner. In the US, passengers could get a compartment which slept one, or two people. I'm assuming the train travel takes place in the late 1800 or after.
Also, when exactly is the train trip? Women did not work on passenger trains until Amtrak was created. The passenger trains had Porters, which were all men and predominantly African-American. Even today, the women are not referred to as a Stewardess.
Let me know if this helps. I have a feeling you will have more questions