No, shouldn't make a difference.  Paddle in the direction the wave is breaking, and you plan on riding.

That said, I'm goofy also, and I have a tendency to paddle on the left side of my board regardless of which direction I'm headed.

But with the shorter boards, I only get a paddle stroke or two on either side before I have to switch, because the board what to yaw so much with each stroke.

So I've also gotten into paddling more parallel to the wave to the point that sometimes I'm getting into it with it at 45^o angle to the wave, and letting it push the nose over in the direction I want to go....if that makes any sense in the written word.

I have a tendency to always paddle in on my frontside as it's just easier to develop power on that side. I'm regular foot so even if it's a left I paddle in frontside then once I'm in I either keep fading right until the wave jacks then get in the backside bottom turn or quickly and get my heel on the backside rail and get truckin' left

Do right handers likes to keep the paddle to the right side?

I'm goofy, right hand, and I like to keep the paddle on the right side, huge disadvantage when dropping in since I have to switch hands while catching a wave. I'm at a beginner stage where once I drop in and make a bottom turn, I completely forget about oar in my hand, so typically for the first half of the ride the oar is facing straight towards the shore not doing anything.

On sup, somethings are easier(like catching waves) but somethings are much more complicated(like wtf do with the oar, shifting weight fwd and bck on the board) than the shortboards.

If by "turn into the wave" he is referring to paddle side only,  I understand your friend's theory, and agree that it may be most advantageous, but most everyone has a "strong side"

I am regular foot. I paddle much stronger with the paddle on my left side, so I almost always set up that way, regardless of wave direction. I simply switch sides after takeoff.

However... if a takeoff is really critical, I much prefer to have my paddle on the "bottom turn" side.... the side towards the shoulder... as your friend suggests. He may be one of the lucky people that are "ambipaddled"
Many times it's the difference between making the wave or getting crushed.

13k - pretty normal when starting out. To bring your paddle into play, try this:
before a wave comes, say to yourself "this time, I use the paddle... regardless of outcome"
Think of the paddle as a longer arm, that allows you to drag your hand across the water as you turn.
By skimming the paddle on the water as you turn, it allows you to lean into the turn a bit more, using the paddle like you use a railing.

I was not switching paddle sides when cutting back, until I began to plan ahead... before the wave came, to make myself do it. Now it is becoming more natural.

Interesting topic. I'm regular foot and paddle right, so I spin left to catch waves. The other day an overhead set was coming in and I was paddling out to meet it before it pitched. Right as I spun quickly left I looked over my shoulder and said something like, "poo-poo"...then I got annihilated. My prone surfing buddy asked me later, "how come you didn't spin right? It would have probably gained you that few feet you needed to make the drop" good point. For some reason my balance gets all discombobulated when I spin right, which is disappointing because I normally have really good balance. Definitely something I'm going to work on with diligence when I get home.

I'm goofy, right hand, and I like to keep the paddle on the right side, huge disadvantage when dropping in since I have to switch hands while catching a wave.

I guess we are creature of habits. Now I can't seem to get into waves to save my life when I paddle with the oar on my right side.

Well it would be nice to be able to do it from either side equally well.
The thing is where do you need to be to start sliding and actually drop in.
Kind of depends on the size of the wave .  If I am facing out looking at the wave
I can turn to the left or right when coming around. If I am in the average takeoff
spot I may have to go in either direction.  I don't like changing paddle sides
when I start to paddle for it . If I have to go to the right and turn around I will be paddling
on the left to get closer to the peak when dropping in if I am in to deep I will tend to paddle
on the right to bring it around. I am much better at paddling to the right and if I am going right
this leaves the paddle on the wave side where I can stick it into or on top of the wave
for control. (assuming a right breaking wave)

Also use a regular foot surf stance 99% of the time.
One of the biggest problems is if you have to paddle for it is
syncing up with the drop in at the same time board is nearly perpendicular
to the wave in most cases though you can drop in a off angles
if the wave is steep enough and of the right size and you are in the right spot..