When does the assistant act on its own?
Today it reads and prioritizes your mail and tells you what needs you — and leaves your mail where it is until you act on it. Triage happens without being asked; touching your mailbox does not.
Having routine mail handled for you quietly — silenced, filed, reported as spam, or archived once it has been dealt with — is a set of behaviors controlled under Account → Automation, and it's part of the Assistant plan, which is coming soon. Silencing a sender, silencing bulk marketing, reporting spam and filing under a reminder each start off; once you turn one on, that configuration is your authorization and it acts on that kind of mail without re-asking, reversibly and with a record in Activity. One thing on that plan starts on, and it does move mail: routine mail the assistant handles for you — newsletters, receipts, notifications that need nothing from you — is archived out of your inbox. It stays in Gmail, in All Mail, and is never deleted; everything archived this way is always shown under Handled for you and in Activity with an Undo, and it has its own switch under Account → Automation ("Archive routine mail I handle for you") — switched off, every message stays in your inbox.
Two more things on that plan start on: an appointment your mail states goes on your calendar, and a job title or company it states for someone already in your contacts gets filled in. Neither moves or hides the message — it stays in your inbox exactly where it was — both are shown to you with an Undo, and both have their own switch under Account → Automation.
The safe default always wins: when in doubt, it asks. Anything sensitive, anything that leaves your account, or anything hard to undo is presented to you as an approval before it happens — no setting turns that off.
See also: How does it decide to ask versus act? · What does it ask before doing?