Users will need Full Write People permission to preform this task.
Users with this permission also have the Limited Read People and Limited Write People permission.

Excluding a Profile from the People Directory
First locate the profile you want to modify.
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On the person's profile > Select the Actions Button in the top right
> Choose "Update Privacy Settings"
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Finally > Mark the Profile as "Unlisted" and save your changes.
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Profiles can also be unlisted by custom criteria via the Directory Settings.
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Other helpful Articles:
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​How do I edit a person's profile? | One Church Software Help Center
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​How do I manage the church directory settings? | One Church Software Help Center

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