My primary carry position is on my "weak" right side, for my weaker right hand to draw. My right hand and arm used to be stronger. (Getting old has its down sides.) One big reason I tend to carry more than one handgun is so my left hand has easy access to a handgun, while my decades of reaching for my right hip in an emergency remains a valid action.
Having said that, I will really complicate things by saying I am, and have always, been a lefty, or more precisely, left-handed. Large, gross motor movements, such as handling large, heavy objects, always felt more natural when performed with the right hand and arm, whereas I favor my left hand for skilled tasks. So, I had a head start in being fairly ambidextrous with firearms. For a several reasons, it seemed wise to carry on the right hip: My first handgun was a 1911, before the day of the ambidextrous safety being commonplace, and when I became a police officer, it was the norm to patrol alone, one to a car, and weapon access was MUCH quicker and easier if the duty sixgun rode on the right hip. As I recall, learning how to draw a heavy duty-type handgun actually felt more natural wth my right hand, which was used to handling heavier objects and tools. (Texas had virtually no legal carry for private citizens back then, so I did not carry until I started wearing a badge.)
Also, decent lefty concealment leather was difficult to find back then, so that further motivated me to go ahead and make my right side the primary carry position.
I now have left and right concealment holsters for the SP101, P229, and assorted medium and medium-large-frame revolvers. If a doc ever finally cuts on my right wrist or shoulder, or if I break a bone or such on my right side, I am prepared to make my left side the primary carry position, temporarily or permanently. If an intrusion occurs while I am sleeping, I will have to deal with a splint being wrapped around my right wrist and hand, and if seconds count, my left hand will likely be my weapon hand. (I favor a long gun for HD, and use my shotguns lefty, anyway, being left-eye dominant; I can run the slide with the splint in place.)