Chromium shows these as one alarming list at install time. Here is what each one is actually for.
The one that looks worst
Access to all sites is the permission people rightly hesitate over. The extension needs it because you can put any site into a session, and it cannot know in advance which sites those will be. What it does with that access is narrow: it replaces the cookie header on requests from tabs you have put into a session. It does not read page contents, and it has no permission to send anything anywhere.
Every permission
host access to all sites
A session can contain any site. Without this the extension could only work on a list fixed at build time.
declarativeNetRequest
The core mechanism. It replaces the Cookie header on requests from a managed tab with that session's cookies. Without it there is no product.
cookies
Reads cookies so they can be copied into a session, and writes a session's cookies back into its own private store.
webRequest
Observes response headers so a cookie set by a site can be folded into the right session rather than the browser's shared store. Observing only; it does not block.
webNavigation
Detects when a tab navigates so the correct session rules are live before the first request leaves, which is what prevents a half-finished login.
scripting
Injects the small script that keeps localStorage and sessionStorage separate per session, and applies the fingerprint posture you chose.
tabs
A session is a property of a tab. The extension needs tab identity to know which session a request belongs to.
tabGroups
Colours tabs by session, which is how you can see which account a tab is in without clicking it.
storage
Stores your sessions and settings on your machine.
unlimitedStorage
A cookie store per session plus the journal exceeds the default five megabyte quota once you have several sessions.
contextMenus
Adds the right-click entry that opens a link or page in a chosen session.
debugger (optional)
Only requested if you enable the deepest isolation tier. It allows header-level control that the ordinary APIs cannot reach. Chromium shows a visible banner on any tab it is used on, and it is off by default.
What is deliberately absent
There is no permission to contact any external server. That is why the extension cannot send your data anywhere: not as a policy, but because the capability was never requested. Telemetry, when you enable it, is the single exception and is covered at what it collects.