Is there a way to play over the local multiplayer, if you are not in the same Wi-Fi? And we can't log in because it's an apk.
No, that will not work. Where should the connection come from?
For that you would need a server.
Yes, I would have one but without registration I can't figure out or is there a possibility?
Buy the game and you can go to a server for illegally downloaded stuff here is no support offered.
To the online zocke. Forcing an Xbox Live account is necessary. For Smartphone and Windows 10 but no additional costs. An Xbox Live profile is Free. The Xbox One and Nintendo Switch, however, looks different - if you want to gamble in Minecraft Bedrock (that's the game today) via Xbox One or Nintendo Switch online, you need an Xbox Live Gold or a Nintendo Online membership.
Local multiplayer gambling is completely free. The same applies to Xbox One and Nintendo Switch - if they are only in the Wlan, but not online, the network does not require membership.
PS: If you have Minecraft as Apk only, and you have not bought it, then you're unlucky. Here, no help to circumvent legal situations is tolerated.
Give yourself a lot of effort and pay if you want something.