When Peggy McIntosh says that recognizing and describing privilege "makes one newly accountable," she means that this recognition confers a duty on the privileged person. This is a duty to use that privilege to benefit others who do not share it by acting as an advocate and an ally. One way to think about this is to consider the question, "Who benefits from my being white/male/rich or otherwise privileged?" If you are the only one who benefits, then you have not held yourself accountable, since you have not used your privilege to improve the position of marginalized people.
Ultimately, one is accountable to oneself to the way one uses privilege. Theoretically, however, one might say that one is accountable to all the marginalized groups in society, and if criticized by members of these groups, one ought to take these criticisms to heart. McIntosh points out that, while you are not accountable to any specific or clearly defined body, you are accountable for helping marginalized people in the way that they tell you they need. Older models of philanthropy involve the privileged deciding among themselves what will benefit marginalized people, and giving it to them whether they like it or not. Instead, according to McIntosh's model, one should listen to and center the needs of marginalized people as they express them, supporting the decisions they make.