The eye is a kind of acute suppurative infection disease between sebaceous glands and meibomian gland. It has no infectiousness. Thus there is no need for you to worry about getting the stye when sitting close to your sister whose eyes get the stye. There are several ways to treat the stye. First of all, it is local fomentation which will promote the fester. And the light inflammation place could get recovery after the local formentation. Secondly, it is incision abscess. Last of all, it is transfusion which could kill all the bacterium. You'd better tell your sister to accept the treatment. And you should not share one towel with her which may bring the bacterium into your eyes.
Yes. In fact, eye stye is just a type of eye infection by somebacteria. And eye style can be contagious. So, it is not a good to get too close to your sister right now though your person's immune system may kill that bacteria before a stye develops on your eyelids. But if you do it, you are still in high risks of get eye stye. So, Just take her to treat her eye stye by an doctor.
Well, generally speaking, you will not get an eye stye from your sister. As we know that a stye is inflammation of the eyelid, usually inside one of the small glands lining the eyelid. Most styes are caused by infection from bacteria. But you should know that they are not contagious, in the sense that you don't "catch" a stye from someone who has one. Styes form as part of a more complicated process than simple contact, and the body's natural defenses tend to block stye formation. And to avoid getting a stye, or transmitting bacteria to another person if you have one, you should pay more attention to regular hand hygiene.