2 breakers, 1 neutral, 14-3 wire coming from Load Center/panel

I'm not an electrician, but know some of the basics. But this is new to me so I want to see what some of you recommend.

Here's the situation (in Iowa by the way)...
Brand new 20x40 shed with brand new Load Center/panel installed inside by professional electrician. Currently has 1 basic light and 1 outlet wired by the electrician. Now I want to add some overhead lights. Using 2 15A breakers, I want to run a 14-3 wire out of the panel into 1 double gang box having 2 light switches where each light switch has it's own breaker and each controls 1 row of lights. So 2 circuits (1 on black and 1 on red), using the same ground and neutral. Each light has six 32W florescent bulbs for a total of 192W and each row has 3 lights so I'll just round up to 600W total per row. I know 15A using 14g wire can handle 1800 watts (1440 max) so the 600W is well under that, but I still want each row of lights to have it's own breaker.

Is this okay, meaning can the red & black hots share the neutral? I thought I read somewhere that this is okay IF the breakers are on opposite sides of the panel. Is it wrong/dangerous to have them on the same side? Does the electrical code speak to how this should be done and if so where specifically is this stated?

Anything else I'm missing? Many thanks in advance.

You can do exactly like you and and but the breakers on the SAME side in the panel. If you want to put it on the other side then you have to make sure its on the other line. Either way you want one breaker on a black line and a breaker on the red line in the panel.Im assuming you ran 240v to your shed. If you only ran 120v there then it doesn't matter where your breakers go in the panel.

thanks alphabet - correct, it's a 240v panel. Okay, I think I was confused. I actually want the breakers on the same side next to each other - not on opposite sides - correct?

Is that because when you install a double pole breaker it's actually using 120v from each side? So if my 2 independent 15a breakers are on the same side and right next to each other, they are actually using the opposite sides anyway - right? Are the sides called "phases"? or something like that? Sorry I don't know the technical words for most of this stuff

The breakers should be on opposite legs (phases) in the panel. If you have a volt meter, you should be able to read 240v across the two breakers when installed correctly. The objective of this is that because they are opposites, the most that the neutral will ever take is the max on either one of the legs. For instance, if there were 12 amps on the black and 7 amps on the red, the neutral should be carrying 5 amps.

The two breakers should be tied together so you cannot turn just one off.