The link below illustrates a pair of circuits in my sister's house. Would these circuits meet NEC if I just move one circuit to a breaker on the other bus, or do I need to isolate the circuits from each other anyway?

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You need to not share the neutral leg. Each branch circuit needs it's own Neutral.

Right now, if you went across the two breakers. With a meter you would see 240 VAC.

That picture is incorret, doesn't expose real problem except the 1 bulb gets its hot from one circuit and neutral from another.
It is not a Multicircuit(2 hots sharing same neutral)as each in photo has its own neutral to neutral Buss. Plus, You should get 240 volts on that test, Not Zero, Every other one Should be Zero.
If you have a red and black sharing a neutral, the 2 breakers should be on opposite phase
(240 Volts between both breakers, if it is zero, you are on the same phase, and if both breakers were drawing 20 Amps, the neutral would carry 40 Amps and be overheated or burn up. Your picture shows separate neutral for each circuit. All you need to do is have last light get neutral from same circuit.

A similar, but different problem I have seen is a 2 circuits connected together, You usually don't find these unless you are turning breakers off and on to find a circuit, but never goes out, till the second breaker is turned off as well, In that case the Hots(#12) could carry as high as 40 Amps.

Yep, the least light bulb on the top row needs to get the neutral from the circuits on the top row. Or get hot and neutral from the bottom row.

donf: Both breakers are connected to the same busbar or phase. As shown in the picture, measuring the hots across the breakers reads 0 VAC.

Stratmando: 1) Read my reply to donf. I may have my terminology wrong, but the picture is correct. I know getting 0 VAC is one problem. Her breaker box may be unusual in that breakers connect to the same busbar in pairs from top to bottom.

Thanks to everyone who responded. I was afraid I might have to get that light fixture's hot and neutral from the same circuit. I was hoping someone wiser would assure me it was OK, even if I didn't understand how. Looks like I'm ripping out wallboard and repainting this weekend.

Madhatter, the photo IS INCORRECT, or it has no 240 Volts to Panel, Terminology has nothing to do with it, 2 breakers next to each will have 240 Volts. Everyother one will be Zero,
Whoever was your electrician was, was an idiot. I can't believe you have to tear drywall up to correct.
Sounds like you are saying the Bulb had a hot and ground from one direction, and a neutral and ground another direction, strangest thing I ever heard.
You won't find someone wiser to tell you it is OK, only someone who does not know will tell you it is OK.
I reread your post to Donf, Picture shows on opposite phase, should be 240, not 0.
Good Luck

Stratmando: Thanks for your continued interest. If you see a multimeter reading 0 VAC, you see what I intended. Her breaker box must be unusual to create so much confusion, and so I will elaborate. The link below illustrates her breaker box. On the right side of the box that I illustrated -- counting from top to bottom -- breakers 1&2 connect to the left bus-bar. Breakers 3&4 connect to the right bus-bar. Breakers 5&6 connect to the left bus-bar, etc. Between 1&2 and again between 3&4.. a raised bar prevents double wide (240V) breakers from attaching there. Those type of breakers can only attach where they draw from both bus-bars like at 2&3 or 4&5.

The light fixtures about which I was asking use only the hot from the downstairs circuit and the neutral and ground from the upstairs circuit. This was done in order to use 14/2 instead of 14/3 between the stairway's 3-way switches. Both insulated wires of the 14/2 are used by the traveler legs of the 3-way switch. I am told that incandescent lights burn out quickly in these fixtures. CF bulbs last longer, but still less than they should. More importantly, I suspected it might be unsafe. Her house is a collection of electrical and plumbing problems that I've never seen elsewhere.

The only way I see to fix this is to tear out enough drywall between the 3-way switches to swap out the 14/2 with 3 conductor wiring. Bummer.

http://www.geindustrial.com/publibrary/checkout/Brochures%7CDEA168B%7Cgeneric

I see! Yes, you would need to install a 3 wire. Currently the install is in violation.

Yet, the others are correct about your drawing.

I also now see, If you want to avoid the destruction, you could use X-10 or other Power Line Carrier devices for control from both locations and use the extra wire for the neutral.
I will provide model numbers and Links for Obtaining Parts if you want to go that route?
Good Luck

For those still confused about the breaker box aspects of my drawing, please look at the pictures below.

Stratmando: I suspected someone wiser might know something that would spare the drywall from my uneducated furor. The X-10, etc. was a great suggestion that I had not considered. But I've already put in the 3 conductor wire, and my sister needed to paint some pre-existing patch jobs anyway. Problem solved. Thanks.

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Madhatter, I did see photo was correct, I Hate those Breakers. They work fine, it's a personal thing.