On Android, many phones (Samsung, Xiaomi, etc.) have a built-in Dual Apps or App Cloner setting that makes a second copy of Messenger, so both stay logged in. iPhone does not allow that natively. Messenger's own account switcher also holds multiple accounts, though only one is active at a time.
Worth noting it can reset after a big update, so keep half an eye on it. – Isla Diaz 9mo ago
Second this. Also make sure notifications are not being throttled by battery saver. – Kofi Lund 8mo ago
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For truly side by side (both visible at once), a desktop unified app lets you add the same app twice as separate accounts. On the phone, Dual Apps is the cleanest for two logged-in Messengers.