I found the solution myself after countless hardware inspections (changed nearly everything). It was not the hardware, it was the software.
Just to make sure that we are talking about the same things:
I posted that I was sure the sending XBee works. It was just that I could not get it on the serial monitor with the help of a receiving XBee mounted on a shield on the and Arduino.
The solution was not to use the two serial statements together, i.e. Serial and SoftwareSerial.
As I removed all the Serial.xxx commands I got my signals from the sending XBee on the monitor.
Serial.begin(9600);
Serial.println("setup");
mySerial.begin(9600);