I'm right-handed but I've had a defective right eye since childhood so I've always shot left since my first Daisy BB gun. When I was a youth, I always shot right-handed rifles & shotguns off my left shoulder because that was what was handed to me by adults.
When I got older, I became really fond of ambidextrous Ruger No 1's (6.5x55, 275 Rigby, 303 British & 9.3x74r) and various 1885's (303 British, 375 H&H, 405 Win & 44 Mag). I have shot deer off of both shoulders with these rifles. The straight-wall 405 Win & 44 Mag are my Arkansas, Louisiana & Mississippi primitive rifles. Ambidextrous falling-block single-shots work really well for the majority of the hunting that I do.
When I started buying bolt actions, I have always bought left-handed models with zero regrets. My rimfires are Browning T-Bolts in 22LR & 17HMR. I have a Cooper, Sako 85, Steyr CL II, Remington 700 BDL, Winchester 70 Safari Express & Ruger M77 MkII in calibers from 22-250 to 458 Win Mag.
I'm not much of a shotgun guy so I just have an ambidextrous bottom-eject Browning BPS. When I do a multi-day goose hunt, I usually shoot the first day left-handed, the second day right & then back to the left for the last day.
I shoot pistols right-handed and left-eyed. I shoot crossbows left-handed and left-eyed. I shoot compound & recurve bows right-handed and right-eyed because my bad right eye is reasonably functional with a peep sight at close range. If I keep hunting with a bow or decide to increase my current moderate shooting distance, I'll probably switch my bow hunting to left-handed and left-eyed.
My only right-handed rifle is my Wilson Combat Protector Carbine. The AR-15 is such a popular rifle I decided to get the much more common right-handed model.
So my first choice is usually ambidextrous & neutral. My second choice is left-hand specific. Right-handed is avoided with one exception that I debated for a long time.