They do use their infrastructure to connect to Apple Notification Service servers when the app is not used, they do act as a Man In The Middle but in a secure, concise manner (not in a sketchy way). And they conect to their servers for registration and subscription status. You can read a more in depth explanation on their blog

https://blog.beeper.com/p/how-beeper-mini-works

Besides all that I'm not sure if someone who wants to create it's own implementation of all of this can do it without any apple device; reading jjtech technical explanation (https://jjtech.dev/reverse-engineering/imessage-explained/) where he explains pypush he mentions the obfuscation process for registering a device to apple servers, here is where pypush somehow manages to convince Apple that the machine is genuine, there is a mention there to some serial identifier stored on a file called data.plist, if someone wants to implement this proof of concept would need to give another serial identifier?