Simple constant current regulator for a 1.6A laser LED
The circuit you designed is fine.

Connecting multiple current generators in parallel can be done and is done in many applications. You need to be sure that the output voltage is within the range each of them can support.

To calculate the power loss on each regulator, you can treat them separately.

Your LED needs 4.6 V, starting from 12 V the drop is 7.4 V. Since the current is half an amp per regulator, each of them will dissipate 3.7 W. This power is dissipated on the regulator, the sense resistor and the diode. Since the resistor is 2.5 Ohm, the drop on it will be 1.25 V, while the diode will drop approximately 0.7 V. The LM317 will see only ~5.45 V of the total 7.4 V drop. Again, multiplying for the current you finally get 2.7 W on each LM317, 0.625 W on the resistors and 0.35 W on the diodes.

The total dissipated power will be 11.1 W. You will need 1W resistors, and heatsinks for the 317s.