No, I jumped pins 2 & 14 (SDI & SDO) on the first register, just to verify that the circuit of the second IC was operational. Which it was, even though it was then operating as the first register.
Oops. My mistake. I was looking at the schematic for the project you pointed to which uses 74595 shift registers. I think the way you had it was right. Have you tried swapping the two registers to make sure both act the same in either location? If so it sounds like the second socket isn't getting one of the signals: MOSI, CLK, or SS.