Switch Input to Arduino Serial to Arduino Serial Output Led High or Low.

You send messages via the serial port. If you have TX of one Arduino connected to RX of the other, and vice versa, with grounds connected, then:

void loop()
{ // Down here where it belongs
  buttonState = digitalRead(buttonPin);
  if (buttonState == HIGH)
  {  
    Serial.print('1');  
  }
  else
  { 
    Serial.print('0');
  }

Then

  int Rx = Serial.read() - '0';
  
  if (Rx == 1)
 {
  Serial.println("Led is on");
 digitalWrite(LedPin,HIGH); 
 }
 
 else if (Rx == 0)
 {
 Serial.println("Led is off");
  digitalWrite(LedPin,LOW);
 }
 
 else
 {
   Serial.println("Invalid!");
 }

But no Serial.flush(). Throwing away random amounts of unread data, or blocking until one character is sent, depending on the version of the IDE you are using, makes no sense.