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.