I'm right handed in most things and have been experimenting with shooting right handed ( with both hands) but using my left eye for sighting. It seems to be working fine.

I'm left eye dominant and it also has the better sight, so it gives a much sharper picture than using my right eye. Is this a common thing to do, or are their some downsides re accuracy?

I am right hand dominate and since I don't see a whole heck of a lot out of my right eye, left eye dominate. So far it has worked for me. lol

Ditto.

While I don't see any particular advantages or disadvantages when shooting handguns, there is a definite advantage when shooting a rifle as "cross-dominant", where your strong hand is controlling the forend of the rifle.

The disadvantage of the left-eyed, right-handed combination is you may be stuck shooting a rifle left-handed, which can limit your options of models you can shoot safely or easily.

Use whichever eye works the best. It's usually a little harder for a left eye dominant person to shoot right handed but if it's working for you, stick with it.

__________________
Minimize the variables

Agree here as I too am right handed but left eye dominant. With handguns, there is no issue; with long guns on the other hand, the long gun must go on the left shoulder as it's easier to train your hands to shoot off your left shoulder, than it is to train your brain to become right eye dominant.

Once upon a time there would have been a myriad of forces trying to make you use your RIGHT eye. Of late, most gun makers leave some wiggle room for whatever the customer

I'm also right handed and left eye dominant.

I do shoot with my right eye for precision though, just fine. But then, I can also swap to RH RE just fine.

My wife just realized she is also left eye dominant and found it's changed her game. She isn't too experienced, so we swapped her to shoot left handed as well. That's working great for her, she just needs to work on strength.

The conclusion I have come to is... no one is the same. The degree can be tiny.

Saying that, the ideal is to shoot the hand your eye matches.

If you shoot cross, do get some help with an instructor to make sure you are not compromising your arm position and grip to make up for it.

Just to throw another wrinkle into the dominant eye issue. My wife is RH with left eye dominant and has a lot of trouble with accuracy. One day I let her shoot my Open pistol with the reflex sight, no accuracy problems. Apparently since only one eye can see the red dot, there is no problem with cross sighting. If the dot is on the target, it hits. Problem with this for me is I had to buy a reflex for the wife. ($$). . I hadn't noticed this since I'm RE/RH and switch back and forth between reflex and iron sights depending on what kind of match I'm shooting. Just throwing this option out there, the reflex sights also helps aging eyes.

__________________
Minimize the variables