Once again, a conversation with a new Morpholino user led to a discussion others might find useful.
For the two non-overlapping oligo specificity control, you do two separate sets of injections of the two oligos targeting the same miRNA. If the embryos treated with the oligo targeting the 5' end of the miRNA produces the same phenotype as the oligo targeting the 3' end of the miRNA, then that is good: it supports the idea that the phenotype you are seeing is caused by knocking down the activity of the miRNA you intend to target, and not caused by binding to an unexpected RNA.
You might think that only the oligo that targets the guide strand of the miRNA would give a phenotype. The reason both of the oligos should work is that Morpholinos can invade the pre-miRNA and pri-miRNA before they have been processed into the mature double-stranded miRNA. Either one of the oligos can invade the immature miRNA hairpin and once the oligo is bound there is no longer a double-stranded RNA to be processed and become the mature miRNA. By opening up the hairpin and displacing part of the RNA, the Morpholino is acting as an inhibitor of miRNA processing enzymes (e.g. Drosha, Dicer). If both oligos can inhibit maturation of the miRNA, loss of the mature miRNA should produce the same phenotype in the embryos.
Kloosterman WP, Lagendijk AK, Ketting RF, Moulton JD, Plasterk RHA. Targeted inhibition of miRNA maturation with morpholinos reveals a role for miR-375 in pancreatic islet development. PLoS Biol. 2007;5(8): e203.