So long as you do not run items from the secondary hard drive, there is very little that is likely to go wrong.
If running Malwarebytes AntiMalware, you might as well run another while the disk is available. I use Superantispyware (but suspect it might not be quite the bees knees that it was).
When the HDD is put back in it's own machine, antispyware/av should be run again; although they will clean a slave drive, they won't catch everything by this method.
As to connection, it won't matter from the point of scanning, although SATA would be the fastest. I assume it's either (SATA/IDE) or (USB)... ie depends what kind of HDD... USB will be slowest.
An alternative scenario would be to boot the "other" computer with something like Ultimate Boot CD for Windows, and run antispyware from that - leaving your HDD completely out of the equation. (... or do you just have the HDD on its own?)