Finding server and instance name
If you're really using LocalDB and haven't just named a regular SQL Server instance as if you were, there are definitely some extra hurdles in establishing connectivity from remote machines. This feature is targeted primarily at local, isolated development.

I wrote a pretty lengthy article on LocalDB, including how to get over some of the connectivity hurdles, in this tip (and there is likely some valuable information in some of the comments, too):


Getting Started with SQL Server 2012 Express LocalDB


More recently, I added a tip to handle a more modern version, which I highly recommend if for no other reason that most of the connectivity hurdles have been reduced or eliminated:


Getting Started with SQL Server 2017 Express LocalDB


I would strongly recommend installing a proper instance of SQL Server (even if only the Express Edition), which can be set to always be running (no matter who is or isn't logged into the machine), does not have any of these additional connectivity hurdles, etc.