The output point, Vout, is connected to that point on the other circuit. The actual output of each op-amp is only connected to its diode. It is not connected to anything else, especially not the other op-amp output.
What will happen is that the inputs will be unequal. The Vout will be the same for both as they are connected together.
The amplifier with the higher input will try to raise the output, and it will be able to because its diode will conduct. The amplifier with the lower input will try to lower the output, but all that will do it turn off its diode, so that amplifier output, but not Vout, will go as low as possible.