We have studied the phase diagram of two capacitively coupled Josephson junction arrays with charging energy, $E_c$, and Josephson coupling energy, $E_J$. Our results are obtained using a path integral Quantum Monte Carlo algorithm. The parameter that quantifies the quantum fluctuations in the i-th array is defined by $alpha_iequiv frac{E_{{c}_i}}{E_{J_i}}$. Depending on the value of $alpha_i$, each independent array may be in the semiclassical or in the quantum regime: We find that thermal fluctuations are important when $alpha lesssim 1.5 $ and the quantum fluctuations dominate when $2.0 lesssim alpha $. We have extensively studied the interplay between vortex and charge dominated individual array phases. The two arrays are coupled via the capacitance $C_{{rm inter}}$ at each site of the lattices. We find a {it reentrant transition} in $Upsilon(T,alpha)$, at low temperatures, when one of the arrays is in the semiclassical limit (i.e. $alpha_{1}=0.5 $) and the quantum array has $2.0 leqalpha_{2} leq 2.5$, for the values considered for the interlayer capacitance. In addition, when $3.0 leq alpha_{2} < 4.0$, and for all the inter-layer couplings considered above, a {it novel} reentrant phase transition occurs in the charge degrees of freedom, i.e. there is a reentrant insulating-conducting transition at low temperatures. We obtain the corresponding phase diagrams and found some features that resemble those seen in experiments with 2D JJA.
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