DB Engine :: Streaming And Table Valued Parameters

I want to insert lots of data into two tables. For this I want to use table valued parameters and a stored procedure. So, what is the better way for best performance? Using two stored procedures or a single procedure with two parameters? How does SQL Server consume the data if I use a single procedure with two parameters. Is it really streaming the data? I mean, is SQL Server already starting to insert the first rows as soon as it gets it even if the client is still sending more and more rows and than the same with the next table?
e.g. I have a procedure like this:

CREATE PROCEDURE [dbo].[usp_UpdateElements] (
@tvpElementsToInsert As [dbo].[tvpElements] Readonly,
@tvpElementValuesToInsert As [dbo].[tvpElementValues] Readonly) AS
                Begin
                    Insert Into Elements Select * From @tvpElementsToInsert;
                    Insert Into ElementValues Select * From @tvpElementValuesToInsert;
                End