Shared SQLLocalDB instance means that a single local instance may be accessed by any other allowed user.

sqllocaldb share <user> <name> <shared-name>

<user> - user name of the owner of the SQLLocalDB instance to share,

<name> - name of the SQLLocalDB instance to share,

<shared-name> - name of the shared SQLLocalDB instance.

Access to local SQLLocalDB instance is made using (localdb)\<name> as data source in connection string.

Access to shared SQLLocalDB instance is made using (localdb)\.\<shared-name> as data source in connection string.

The last thing to do is to create login with valid role for the user.

create login [<domain>\<user>] from windows;
exec sp_addsrvrolemember N'<domain>\<user>', sysadmin

After that user should be able to access shared SQLLocalDB instace with integrated authentication.