Each org is a separate workspace with its own roster, money, roles and records — the isolation between them is enforced at the database, not just hidden in the interface. Your permissions are per-org too: you can be president of one and an ordinary member of another, and neither leaks into the other.
Your roster place is per-org as well. You hold a separate entry in each organization you belong to, with its own dues balance, attendance, GPA, service hours and office — and its own name, so you can be Rob in one chapter and Robert Chen in another. Nothing an officer records about you in one org is visible to, or editable from, another.