Interesting question, and grounds for at least one critical essay comparing Bella Swann (of the TWILIGHT books) to Katniss.Both characters are very self-sacrificing, but I think Katniss is more sympathetic because Collins is a more experienced writer and has learned how to get the information that she is self-sacrificing across to the reader successfully, where TWILIGHT was Meyer's first book, and she didn't really know how to get the information across (she has improved, and she did a better job with her also-self-sacrificing character, Wanderer, in THE HOST).
It's very hard for a first-person narrator to come across as self-sacrificing because it's a trait that requires humility and not thinking about oneself, so the first-person narrator can't just say, "I'm very self-sacrificing." It has to come through in more subtle ways from what the character does and how she thinks.
Katniss risks her life every day to get food for her mother and sister, and the way she thinks about her sister makes it clear that she would do anything for her. And when she does, it's believable, and makes the reader care.
Bella Swann comes across to some readers (or so I've heard--she didn't come across that way to me) as whiny instead. I didn't realize she was self-sacrificing, though, until I read THE HOST and saw that Wanderer was self-sacrificing and realized that Bella was, too.
Anyway, that's my take on it. And one of the reasons why first person narrators can be very hard to get right.