Lineage doesn't have push notifications. You need to use an apps built in notifications or ntfy (if the app supports it)

Some apps such as Signal implement their own push implementations that you can opt to use and they usually suck; unreliable, huge battery drain or both.

Some apps implement support for Unified push which works great IME.

Most apps rely on Google though and it sucks. Some of the risk can be mitigated by using µG rather than the regular Google surveillance services on your device. Push notifications work rather well but it's not without compromise in other areas.