There are some community ordinances and homeowner association by-laws that stipulate such things. I think about it more with something like a stockade fence that has a smoother side and one with horizontal pieces. Something like a stockade fence is climbable on one side and non-climbable on the other, so in something like a pool enclosure it becomes both a safety and legal issue (in my state, MA, code for a private swimming pool is a minimum of 4' high non-climbable.
I'm not sure if the fence in your picture serves a greater function (looks like something to keep horses in), but it may be so that the rails can not be pushed out by livestock. That may be the origin for that fence design whether or not it applies in this use. If that is the case it would be the norm to see the posts on the outside. I don't find that fence less attractive from one side or the other. It is not an aesthetic often seen in my region.
I really like the "good neighbor" privacy fence with alternating panels. That way, nobody gets the "good side". Lol. Why they don't just put pickets on both sides is beyond me. If it's only adding about $200 on a $400K house to have a decent fence, I can't think the cost is too prohibitive.
I think your are in the minority on which way the fence faces these days, the proof is the in the deed restrictions and ordinances that have to be put in place. I agree with you, seeing the structure of the fence makes me shake my head sometimes.