Mystery current on neutral wires coming into outlet box
First, there is such a thing as a multi-wire branch circuit (MWBC), in which two hots intentionally and consensually shate a neutral.  They are placed on opposite poles of service, which means only differential current flows through the neutral.  This works for 2 hots in single (split) phase service, and up to 3 hots in 3-phase (208V) service which you will only find commercially or in NYC. 

Neutrals must be carefully wired to be monogamous to its partner hot(s), to assure that the neutral carries no more current than its partner hot(s).  After all, neutrals don't have breakers.  If a neutral carried current for more than its intended hots, it could overload, and nothing would detect this.  Borrowing a neutral is not allowed.  

Another rule is that currents must be equal in every cable or conduit. This means no loops, wiring routes must be a strict "tree" topology. It also means the neutral will always be in the same cable with its partner hot, or hots in MWBC. 

And that means MWBCs will be readily apparent in the service panel, because they will enter the box in a red-black-white cable.  (or in the same conduit).  When you see that, place both red and black gots on a 2-pole breaker, so that when you switch off one side, the other is also switched off.  That satisfies a basic rule of MWBC. 

At this point, shutting off that breaker must arrest all current flow in the neutral. 

If it does not, then you have a "promiscuous neutral" problem that you should aggressively eliminate. 

Grounds, on the other hand, are allowed to be promiscuous, since current only flows on them during fault conditions.