You can make multiple google.script.run.functionName() calls.
google.script.run.fnk_One()
google.script.run.fnk_Two();//This line of code doesn't wait for the previous line of code to complete the run on the server
Server function two will start running regardless of whether function one has completed or not.
Both server functions will be running at the same time.
You don't need a success handler if there is no need for the client side to process a return value.
Or either one of the client side calls could have a success handler.
Does that answer your question?