From the point of view of security, if it's the type of fence made from upright planks fixed to posts with horizontal bearers, these provide an easy means of climbing over the fence, and so should be on the inside, thus presenting the sheer wall of the planks on the outside to any would-be intruder.
when we replaced one of our boundary fences we just did it the same way as the old one was (bad side facing us). we shared the cost with the neighbour who shares the fence, so thought that was the fairest way of doing it. the fence on the other side has the good side facing us.
i wasn't aware that either side 'owns' a boundary, i thought it was shared.
also, if i was the only person paying for the fence, then i would put it the way round i wanted it i.e. good side facing me.