A Steiner triple system, STS$(v)$, is a family of $3$-subsets (blocks) of a set of $v$ elements such that any two elements occur together in precisely one block. A collection of triples consisting of two copies of each block of an STS is called a duplicated Steiner triple system, DSTS. A resolvable (or near resolvable) DSTS is called self-orthogonal if every pair of distinct classes in the resolution has at most one block in common. We provide several methods to construct self-orthogonal near resolvable DSTS and settle the existence of such designs for all values of $v$ with only four possible exceptions. This addresses a recent question of Bryant, Davies and Neubecker.