You can check if your character owns the weapon by printing it.
I highly recommend you to watch and understand the epic videos on youtube about replication. Zak explains in a very nice overview how to handle what you want in replication.
There’s lot more videos on youtube about it, too.
We all know multiplayer logic is not that simple, but you need to understand RPC logic to start solving these problems.
One simple way that might solve your problem is to Use Switch has authority inside your shooting function.
Create a Custom Event: Server ShootingFunction, for instance.Set it to run on server.
Connect it, right after, to your ShootingFunction.
Inside your shooting function:
Switch has authority -> Authority -> Call Shooting Function
-> Remote -> Call Server ShootingFunction Event -> Call Shooting Function (client should play the effects too)