On one leg or two legs?

Generally, you should install them on one leg. The reason is that both legs in the panel may not be the same voltage. One could be, say, 120V and the other 118V. If you put one circuit on each, then your equipment will have a 2V potential between them when plugged in. This could cause a ground loop through the interconnects, the equipment, the power cord and the wiring back to the panel. If you place the two circuits on one leg (matters not which) then the potential difference will always be zero. However, the length of both circuit wires to the breakers should be the same because a difference in resistance resulting from unequal lengths (ohms per foot) also causes a potental difference.

I would suggest running isolated ground circuits. Each circuit has it's own ground run back to the panel to a separate ground bar. Then the Isolated ground bar has it's own ground run to it's own ground rod. If you are running two circuits, I think it would be worth the small added expense. I am not an electrician, but I play one on tv... nah, but my friend is, and we just did this, and I don't think it would matter if they are on different legs as the voltage is determined from the service wires. but if you have room on the panel, just use the same leg? If you have a meter you can check the voltage.

Why not try it on the same leg and then different legs and report back what you hear?
IMO; I agree with Gs5556 and he is a professional.

if two lines are in use... or ever how many... and all are dedicated to the system...

Use the same phase.

Using differring phases (sides of the breaker box) intorduces possible issues, with one right off being a possible problem.

Each phase in that box is going to out of phase from one another by at least 90 degrees. That's more important than a possible shortfall in voltage variances.

The nearly same thing applies to running a separate ground... as the difference in potential between the two grounds being employed can be an adverse effect... if only by a few volts... and then there's the issue of creating a possible ground loop too.

the issue rests with the neutral or common (white) leg of the ckt. That sole item is constant regardless what you use or how you employ dedicated lines unless you run an entirely separate service.

These above listed considerations can range from insignificant to quite detrimental and most audible artifacts.

The very best scenario past a new service is the above post on employing transformer isolated dedicated lines.

Due to the common wire being constant throughout however, power conditioning or filtering might prove out as a more suitable solution for you... if none is/are in place presently.

I have four dedicated lines. I have four power filters as well. All my ded lines are on the lighting phase, which works for me best, as I've no one here but myself. I felt it better than to place them onto the appliance side and avoid the fridge, hot water heaters, washer dryer etc.

Ded lines are a plus. They may or may not be the entire answer.

Good luck.

Hanaleimike - While your idea of running separate grounds all the way to isolated/separate grounding rod sounds interesting, it is both dangerous and will likely violate your local building code. It's also not legal to work on your own electric panel without a license, although many ignore that.

Wiring a home with two isolated ground rods may also void your homeowner's insurance should you have any sort of electrical problem.