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In a nutshell, a proxy group is a list that holds service accounts for one or more users. This is used by FireCloud for managing access permissions to project data.
Each user is provided with a unique proxy group, displayed in the user's profile page in FireCloud. This proxy group contains a user's login identity, alongside any service accounts FireCloud may make on their behalf.
As a FireCloud user, you should generally not need to think about your proxy group. You can share workspaces and the data they contain within FireCloud with any other user based on their login identity. However, if you want to share non-public data that is stored in buckets NOT associated with FireCloud workspace for the purpose of running FireCloud workflows, you must use the recipient's proxy group instead of their login username.