I'm going to tell my mother my sister got a second job
Why would you tell her that?
She doesn't pay for anything and always claims she's broke.
If she doesn't want your mom to know you shouldn't tell. It's a secret and she expected you to keep it.
My mom will find out anyway, since my sister was talking loud to her friends about it.
Well, it's fine if your mom finds out on her own. But, if your sister doesn't want to tell her, that's her business not yours.