As Olaf says, this is not going to work out, if for slightly different reasons.
Yes, you can put the local DB on a network share, so that a user on a different machine can access the database. But localdb runs in the user address space, so if you then try to access the database by running LocalDB on your instance, the database is locked by the other instance.
And, no, you cannot access a localdb instance running on a different machine. Again, localdb runs in user space.
So Olaf is right: you need to run the full SQL Server Express edition, and configure it to accept remote connections.