I have a Belkin Wemo switch that requires a neutral wire for installation. There is no neutral in the switch box. However, there is a neutral in the electrical outlet box below it, and a path from the switch to the outlet. Can I run a neutral wire from the switch to the neutral in the outlet box and connect it as a shared neutral? I guess this is another way of asking of all neutral wires are basically interconnected eventually (leading back to the main electrical box) and whether any neutral in the house can be used by anything requiring a neutral.
Another reason not to do it, is because you could end up overloading the grounded (neutral) conductor. Depending on if the two circuits are on the same branch circuit, different branch circuits on the same leg, or different branch circuits on different legs, you'll run into different situations. However, in two out of three of these situations, it is very easy to overload the grounded (neutral) conductor.
Let's examine what happens in a 120/240V single split-phase system, in a few different configurations. The circuit breakers in these examples is a combination circuit breaker, with 15 ampere thermal overcurrent protection and magnetic short-circuit protection.
Load on a circuit
With a single branch circuit supplying a 15 ampere load, we see that there will be 15 amperes on the grounded (neutral) conductor. This is fine, since the grounded (neutral), and ungrounded (hot) conductors are both sized to carry 15 amperes.
Two loads on the same branch circuit
In this situation, we'll see 30 amperes on the grounded (neutral). However, since we're also drawing 30 amperes through a single breaker with 15 ampere overcurrent protection, the breaker will open.
Two loads on the same leg, but different breakers
This is the dangerous situation, since the breakers will not trip. Each circuit will draw 15 amperes on the ungrounded (hot) conductor, but the grounded (neutral) conductor will see 30 amperes. Since the conductor is only rated for 15 amperes, the conductor is in danger of overheating and starting a fire.
Two loads on different legs
If the two loads are on different legs. You'll find that while each ungrounded (hot) conductor sees 15 amperes, the grounded (neutral) carries 0 amperes.
Net necessarily, if the neutral is on a different circuit. First, if either circuit is protected by a GFCI circuit breaker, it will trip. GFCI breakers detect a difference in current flowing on the hot and neutral wires. Normally, this difference would represent current flowing to ground but in this case, it would be current flowing to the other circuit.
The other issue is of load. You are adding load to a neutral wire witch may already be carrying it's full rated load.
I suspect that the device you're adding draws minimal current on its own so issue #2 is probably not a practical issue, though GFCI breakers would still be an issue. Never the less, it may not be to code, though I cannot quote code to prove it.
(Answer assumes you are in the US with a house relatively up to code)
In this case, yes.
Ultimately, all Neutral wires do come back together in the electrical panel box to answer that question as well.
However, while very close GROUND and NEUTRAL are not the same and should not be swapped so make sure you're tapping a neutral and not ground if neutral is what you need.