Two multimeters in continuity mode, only 1 buzzer
'Continuity Mode' is not a well defined function. It is actually a resistance measurement, set to beep for resistance below some threshold, and silence above that threshold. 

In making a resistance measurement, a meter sends out a small current, 1mA would not be surprising, and measures the voltage across the probes. If the continuity reading used a 100ohm threshold, then it would beep for &lt;100mV measured across the probes, and stay silent above 100mV.

When you connect two meters together, you can assume that their resistance measuring currents will not be identical, although nominally the same. The higher current one is likely to overpower the low current one, resulting in a measured voltage that one sees as sub-threshold, the other as super-threshhold.

Try putting one on continuity, the other on current, then swap. Also, get a third meter, and measure the probe voltages when your two continuity meters are connected.