Let me redraw your schematics. (In fact, you should get into the regular practice of redrawing any schematic you don't feel you understand well using well-understood rules.)
simulate this circuit – Schematic created using CircuitLab
Current flow should be arranged so that the top of a schematic is the most positive and the bottom of the schematic is the most negative. Signal, if applicable, should flow from left to right, with inputs on the left and outputs on the right.
Don't bus voltage rails (or ground) around. You don't need to see all the connections as it doesn't matter (mostly) for understanding a circuit. I've eliminated the useless wiring you added. Pointless. Just label the nodes where you know the voltage.
This will save you a lot of grief.
Finally, it's just fine to swap series-arranged, two-terminal devices. I swapped your diode/resistor series chains to put the diodes closest to the positive node voltages to make it a little easier to understand. I think you should also be able to see that they are both on -- or, at least, that you cannot find an a priori reason why either one of them should be off.
At this point you should be able to work out the voltage for \$V_X\$ and, from that, all the currents involved as well. Then you can go back and verify in your own mind the on/off status of each diode, if you like.