In the reverse-biased direction, if your meter reads a resistance value at the small amount of leakage current, you will get a resistance reading, but it is not very meaningful. To test diodes in a more certain fashion, rig a voltage source in series with a fairly high resistance, say 9V in series with 1 Kohm resistor. Then use your meter in current mode putting the diode in series with everything else, so you have a lopp consiting of the meter, the voltage source, the resistor and the diode. I see that a 50V 1A diode has a 10 microamp current when reverse-biased, at rated voltage. That one is rated at a maximum of 200 microamps reverse-biased.
Diodes have two failure modes.
In the other, the junction separates, and they become a non-conductor. But it is necessary to be sure that you have sufficient voltage to forward-bias the diode. If you use the method I recommended above on these devices, in the reverse-biased direction they would be subjected to 9V, and would be destroyed. THey conduct in one direction, with a consistent difference in voltage. The voltage between the terminals for a silicon diode is about 0.6V, for a germanium diode is about 0.3V, and for LEDs is 1.2 to 1.4 or more, depending on the substance from which is is made.