It really depends on how the designer has implemented it, and works on a module by module, and algorithm by algorithm basis...
you really need to read the manual for the specific module
usually the left input doubles as a mono input... some stereo algorithms on some modules (often reverbs) will only take a mono input (and some will use the right input for something else - clocking for example)
some modules will output a signal only on the left output, others will just make a copy of the mono input and send it to the right output (this potentially means that they work as a dual mono output - the same signal is sent to both the L & R outputs) and others will 'stereoise' the mono input (usually by adding a slight delay) - so that the diffeent signals are output from the L & R channels...
"some of the best base-level info to remember can be found in Jim's sigfile" @Lugia
Utility modules are the dull polish that makes the shiny modules actually shine!!!
sound sources < sound modifiers < modulation sources < utilities