No, you should not copy the wsdl to the client and run it.

In your original post you said you are using wsdl to make the proxy-file to
include in the client app.  I suppose this means you are using   wsdl.exe
on a wsdl file.   Once you generate the proxy from this step, you need to
compile it.  The output of that is a DLL.  This is one of the pieces that
needs to be installed on the client.  The DLL, not the wsdl, nor wsdl.exe.

There are 2 stages we are discussing here:
 design and compilation - in which you need wsdl.exe, csc.exe, nmake.exe,
and so on.  You need the sdk for this stage.  The SDK also requires the
redistributable!
 running the app - in this stage you need the "redistributable", which is
the runtime piece.  You don't need the SDK.