Briar has no server: forum posts travel directly between phones, over Tor. The bot is a member of every forum it lists and stays online, so it passes posts on between members. Your phone only receives new posts, and sends yours, while Briar is running and connected to the bot.
Posts written while you are away wait on the bot and arrive when Briar connects again. Yours wait on your phone.
Android stops apps in the background to save battery, and Briar then stops receiving anything.
Sign Out stops Briar completely. Signing out is never needed to close the app: leave it signed in.
After the phone restarts, Briar stays stopped until you sign in again. Turn on Settings → Notifications → “Remind me to sign in” to be reminded.
Not for forums: the bot is always online, so your phone catches up as soon as it connects. A Briar Mailbox mostly helps private messages with contacts who are often offline themselves.
Send the bot /meet 23 to meet another member of forum number 23 who asks too. Once you are
contacts, share the forum with each other (Share Forum in the forum's menu): it then keeps syncing
between you even when the bot is offline.