The manga simplifies things a lot, when some stuff is explained later some is not.
Roast was a mercenary, she lost a eye and arm and therefore her ability to fight, most likely the country they are in have taxed and who doesn't pay becomes a slave, her past is explained later-
But Chikin is not, she was used by a alchemsit, her internal organ were taken away and then the alchemist made some half assed measure to make so she wouldn't die in days, then sold her, hence he tricked the slaver and I wish we would have got achpater with MC telling what had happened and the guy being arrested, sadly is totally skipped, Chikin wasn't diying of a illness but her health body not being able to function anymore.
Mince...is not explained what her problem is, she seems "whole" in the slave shop, and is even explained that her race have strong regeneration, it just says she have a problem in her head and she ia former soldier, so maybe she was enslaved for not doing things like the army wanted.