Spin-orbit coupling (SOC) plays a crucial role in magnetic and electronic properties of 5$d$ iridates. In this paper we have experimentally investigated the structural and physical properties of a series of Ir-based double perovskite compounds Pr$_{2-x}$Sr$_x$MgIrO$_6$ ($x$ = 0, 0.5, 1; hereafter abbreviated as PMIO, PSMIO1505, and PSMIO). Interestingly, these compounds have recently been proposed to undergo a transition from the spin-orbit-coupled Mott insulating phase at $x$ = 0 to the elusive half-metallic antiferromagnetic (HMAFM) state with Sr doping at $x$ = 1. However, our detailed magnetic and electrical measurements refute any kind of HMAFM possibility in either of the doped samples. In addition, we establish that within these Pr$_{2-x}$Sr$_x$MgIrO$_6$ double perovskites, changes in Ir-oxidation states (4+ for PMIO to 5+ for PSMIO via mixed 4+/5+ for PSMIO1505) lead to markedly different magnetic behaviors. While SOC on Ir is at the root of the observed insulating behaviors for all three samples, the correlated magnetic properties of these three compounds develop entirely due to the contribution from local Ir moments. Additionally, the magnetic Pr$^{3+}$ (4$f^2$) ions, instead of showing any kind of ordering, only contribute to the total paramagnetic moment. It is seen that the PrSrMgIrO$_6$ sample does not order down to 2 K despite antiferromagnetic interactions. But, the $d^5$ iridate Pr$_2$MgIrO$_6$ shows a sharp antiferromagnetic (AFM) transition at around 14 K, and in the mixed valent Pr$_{1.5}$Sr$_{0.5}$MgIrO$_6$ sample the AFM transition is shifted to a much lower temperature ($sim$ 6 K) due to weakening of the AFM exchange.
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