We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $α\in(0,1)$ and with idiosyncratic Poisson innovations. Assuming that $α$ has a density function of the form $ψ(x)(1 - x)^β$, $x\in(0,1)$, with $\lim_{x\uparrow 1}ψ(x) = ψ_1 \in(0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $β\in(-1,0)$, $β= 0$, $β\in(0,1)$ or $β\in(1,\infty)$, when taking first the limit as $N\to\infty$ and then the time scale $n\to\infty$, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaite and Surgailis (2014) by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.