I have seen long guns (mainly double shotguns/rifles and drillings) made in England or Europe with offset stocks, to allow a right side hold with a left dominant eye. But unless you want to take out a second on your house, and sell your youngest daughter into lifelone servitude, that probably isn't an option for you. So I would suggest basically what the others have, either shoot with the "weak side," (as a former longtime law enforcement weapons instructor, I have always hated that term), or learn to ignore the dominant eye and shoot with the other...

I am left eye dominant and left handed. I have been shooting all types of right handed long guns all my life. Ejected brass, shells, etc is just not that big of an issue. My advice? Practice, practice, practice, shooting left handed, Peroid.
Shooting across the gun is never the answer, and it just looks stupid. Guns are not designed to be aimed this way. I only have one left handed firearm (a bolt action) and quite frankley I don't really see a big advantage.
So shoot the righty guns from the lefty position and you will be fine.

To which you've hit on the key to success my friend.

Good shooters practice a lot. There are those who have a knack, plenty of natural talent, but they lack the experience to be consistent for the most part.

If you are willing to put in the hours and the resources to develop proficiency, you will be rewarded with success. To that end, practice doesn't mean just shooting from the same position over and over. Quite the contrary, any seasoned expert will practice shooting from different positions, different sides, etc. Then the expert can take that awkward shot and reasonably expect it to go where he intended it to. Familiarity with the firearm of choice proves marksmanship; familiarity with varying techniques demonstrates the difference between having a shooting solution or walking home empty handed, or worse.

I'd like to point out that if you're planning on using a firearm for personal defense, there's a lot more to learn than just shooting. Practice should include drawing (or unslinging if we're talking about a longarm), movement, using cover, reloading, and clearing jams. Alertness and verbal disengagement skills are important, since it's better to avoid trouble in the first place. Of course, learning to make good shoot/no shoot decisions is important if you want to stay out of prison.

Shooting is an important part of defensive firearm use, but it's only one piece of the puzzle.

While not an expert, I do have a question. What do you want to use a rifle for? Is it for hunting, formal target shooting, plinking, etc., because your reason for the rifle vs a handgun will slant the answer. If it's for hunting, may I suggest a TC encore or contender, or one of the no longer made Savage strikers, or other hand cannons. I used to have a striker in .308 and used it deer hunting, until I forgot to close the muzzle brake and the noise physically hurt. I decided to sell it, but it was very accurate even at long range (with a rest.) It really comes down to what you want the rife for.

Take Care and Stay Safe.