This is a fundamental question but for some reason I always get turned around.

It would seem to me that the pitcher should have his arm on the inside (RHP on first base side of rubber with arm in the middle and LHP on the third base side of rubber with arm in the middle).

Is this the correct mechanic and if not, what is the logic?

LHP son currently on the first base side and it looks like he's throwing across his body and is bothering me...

I'm an LHP as well and I throw from the 3b side of the rubber. Not extreme, only my heel is off the rubber when coming set in the stretch, but definitely towards the 3b side. The original reason I started throwing from that side was because I found it easier to locate the ball on both sides of the plate. Since (in general), a lefty's ball has natural tail on it, I agree and I found it easier to throw strikes to both sides of the plate when the arm was "in the middle" of the rubber. I don't know if there is a proper method to it though. To each his own I suppose.

It depends on what you are trying to do. If you are trying to take advantage of a tailing fastball then RHP on the first base side would be typical. If you are trying to get a better break on the curve then the other side would usually work best.

In the end, it is whatever works best for the individual pitcher and that sometimes is a matter of comfort more than anything else.