I'm sorry, but I'm afraid some details from this point forward will be a bit lacking from me, simply because I'm not an application coder.

To the basics: An executable called from a particular machine will run in the memory/cpu space of whatever machine called it. This is normal.

You want to run an executable from a machine that's not originating the call. That'd be Server 2. The only way to do that is to have a service (rt-click My computer, Manage, services) that will accept foreign input and then will activate a local .exe. Whatever is the login for SQL Server on Server1 will need rights to be able to send a message to the service on Server2, and that service will then fire the .exe call.

I would recommend speaking with your app developers on this. It shouldn't be overly difficult for them to create an active service for foreign requests, but it may break internal security policy. Either way, they'll be able to better answer what's the acceptable solution in your company far better than we will.