Using global nonlinear optimization one can obtain a configuration of $19$ spheres, that touch at least one of the central unit spheres and have almost no overlap. In fact, if one takes their radii to be $.99$ instead of $1$ they are non-overlapping.
Below are the coordinates; the two central spheres have radius $1$ and are centered around the origin and $(2,0,0)$. Here is a picture of the configuration:
Maybe this is helpful as a starting point in the simulated annealing approach you mentioned in the comments, but I am not so sure, since it seems to be somewhat jammed already.
(1.30155675907051, 1.87408031823623, 0.000000000000000),
(3.30307693251716, -1.48756032724292, 0.298587978254738),
(3.77087392448039, -0.00565125397965555, -0.929501805767173),
(2.34028624585583, 1.21452857880052, -1.55213581949477),
(1.49324421375722, -1.89375136244257, -0.396111537780328),
(3.31658479709791, 0.0846873881998760, 1.50314088439273),
(1.46434039083497, 0.727511022954233, 1.78431961671711),
(1.82006459231285, -1.21042374283863, 1.58192844712867),
(2.17723615440802, -0.736686335972064, -1.85091344691338),
(3.24812939881743, 1.55286888817322, 0.175417273814499),
(-0.234051719598306, 1.57207701538230, 1.21399903223316),
(-0.182142247556194, -1.45688908753032, -1.35804947932633),
(-1.35706161319061, 0.134224681025498, -1.46299949180272),
(-1.82011109002214, 0.745105549605677, 0.363336400498419),
(0.560694305152308, 0.407540066521993, -1.87604184131108),
(-0.536544075078242, 1.78215924389985, -0.732139935326408),
(-1.60854779879003, -1.18847287273622, -0.0103058129362229),
(-0.850339283181967, -0.217696133065399, 1.79708972984823),
(0.0325533674370459, -1.76736000103794, 0.935616858014407)]