You are in for some trouble, since your existing wiring is completely
against code, and a bad situation to boot.
Just think for a second about how this circuit looks. The current's
path starts at a circuit breaker (on a black wire) goes to a switch 1,
goes to switch 2, goes to the outlet, goes into the appliance, (and now
on the white wire) back out of the appliance, back to the outlet, back
to switch 2, but now takes a turn and heads down a different path to
possibly a handful of other boxes and junctions, who knows where else,
then eventually back to the neutral in the breaker panel.
Essentially you have a big 1-wind electro-magnet here. Besides breaking
code, I can think of dozens of consequences, e.g.:
You are broadcasting emf interference all over the place, probably
interfering with radios, televisions, and other wireless devices. (And
I have heard stories of such circuits interfering with hearing aids and
all sorts of other sensitive electronics and things with antennaes,
even over to your neighbors houses).
You can get inductive heating in several of those junction boxes,
posing a fire hazard.
Heaven forbid someone reverse the hot/neutral in some other place in
your house that happens to share one of those power sources. You would
either get two same hots at your outlet, or 240V. You don't even
mention if the two power sources are on the same circuit breaker or
If someone puts a GFCI somewhere on one of those two now-joined
circuites, it would either trip constantly (b/c of too little current
on the neutral in the one case), or it would not protect you in the way
it is supposed to (b/c of too much current on the neutral).
In other words, besides the potential for future changes to make this
much worse than it already is (and who knows, maybe it already is much
worse -- you don't know unless you have traced through the WHOLE
circuit on both sides, everywhere it goes), you have some active
problems like inductive heating and radio interference.
Lucky for you, this is pretty easy to fix. Just ditch the one power
source, and add a 12-3 cable where it belongs.