The person getting shocked is essentially a resistor in parallel with the load. Both the load's resistance and the person's resistance will have the same voltage applied. Therefore, the amount of current going through the person's body will not have anything to do with the amount of current going through the load. The current in any one of two parallel resistors depends only on the value of its resistance and the value of the applied voltage.
In the discussion above, I have disregarded the very, very small change that takes place inside the voltage source, each time new load is added to the circuit. That change is a reduction in the output voltage of the source, and takes place because of internal resistance within the source. The higher the original load current (i.e., prior to the person getting shocked), the higher the current will be through the source, and thus the greater the voltage drop internal to the source, the lower the value of voltage at the terminals at the source, and the lower the voltage will be when the person touches the circuit. Therefore, if you take into account small changes inside the source, your friend's statement is correct (i.e., the shock does vary with the amount of the original load). However, the difference will not be enough of a reduction to change the person's chances of survival.