Chrome extensions [1], and Firefox WebExtensions [2], can have background scripts that can continue running even when the browser window is closed. So in theory, with a p2p filesharing extension, you may not have to keep any browser window open or even minimized.

[1] https://developer.chrome.com/extensions/background_pages

[2] https://developer.mozilla.org/en-US/Add-ons/WebExtensions/An...