Is it possible to get two separate programs to communicate on the same computer (one-way only) over UDP through localhost/127... by sharing the same port #?
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My advice: don't pass the port number into the UdpClient constructor. From the documentation, (somewhat sparse, I know...) it looks like if you do, the UdpClient will try to bind to that port (which, as sysrqb mentioned, is not allowed). (If you don't, I believe the UdpClient will listen on a random port for any replies. You could also pick a port you know to be unused.)
When you call Connect() you need to pass in the port number the server is listening on.