Big Clive in a recent teardown video looked at an LED light that had a zener diode across each LED of a series string. He surmized they were there so an open LED wouldn't kill the whole string.
If you're using a constant-current supply, you don't need a sense resistor. Voltage across the LED string will be equal to the sum of the forward voltages. If an LED shorts, the voltage will drop by the forward voltage. Voltage across the string would tell you how many LEDs are shorted. If any LED is open (assuming no zener shunt), voltage across the string will be equal to the maximum voltage output of the constant-current supply.