Adding a local user account into SQL Server
This has nothing to do with SQL Server, is just basic Windows authentication. In a workgroup you can only authenticate using NTLM mirrored accounts. Mirrored accounts are local accounts that have identical name and and password on different machines.

So to use Windows authentication between a process running on machine A and a process running on machine B, in a workgroup, you must:


create a local user A\foo on A
create a mirrored user B\foo, on B, that has the same password as A\foo
run the client process as A\foo on A
grant permissions to B\foo on B


I strongly suggest you deploy a proper domain and use AD instead.