Winterbourne is concerned by Daisy's unwillingness to conform to European standards of behavior. The narrator uses an idiom to describe the consequences of her social defiance, as perceived by Winterbourne.
[...] they intimated that they desired to express to observant Europeans the great truth that, though Miss Daisy Miller was a young American lady, her behavior was not representative—was regarded by her compatriots as abnormal. Winterbourne wondered how she felt about all the cold shoulders that were turned towards her, and sometimes it annoyed him to suspect that she did not feel at all.
Giving someone the cold shoulder is an idiom that expresses rejection. James turns it plural to indicate that more than just one person harbors unfriendly feelings towards Daisy. When Winterbourne thinks about "all the cold shoulders" that are turned towards her, he is thinking about the collective disdain that American expatriates in Rome have for the young American. He wonders if she recognizes the effect that her behavior has on her chance of being accepted by the group.
He later uses the same idiom directly in dialogue with Daisy, warning her that the American expatriates will give her the cold shoulder for "going round too much" with Giovanelli. When used in this context, the idiom serves the purpose of softening the blow. Instead of telling her outright that people dislike her, judge her, or have lost their respect for her, the figurative phrase allows Winterbourne to be honest without being brutal.
Winterbourne's warning about the cold shoulder seems to affect Daisy, who hitherto has told herself that no one genuinely cares what she gets up to. This response suggests that she hadn't entirely registered the chill in people's esteem for her. However, Daisy isn't affected enough to change her behavior.
