A good silicon diode will measure around 0.6V, when forward biased and be open circuit, when reverse biased. A bad diode will most likely read short circuit, in both directions, but could fail open circuit in both directions.
Shotkky diodes have a much lower voltage drop, normally around 0.3V and a higher leakage current. High current, low voltage devices might read a low resistance/voltage in both directions, using a multimeter, because the leakage current is relatively high. A more reliable test is to use a small fillament lamp, say 100mA, 6V and a 6V power supply, or battery. The lamp should only light, when the diode is forward biased.