Perhaps the "no drip" side has the water settling in the exhaust pipe or muffler due to an uneven surface.
Stand back, it's about to have a baby in the very near future The exact same thing happened to my wife
Drive it 'till it's warmed up and see if it still does it. Only one side is a bit concerning but maybe one side drains and the other doesn't. I suppose you may have a bad head gasket or a cracked head (or worse), but it's too early to tell for sure and tough to know from here.
If it has no smell and clear drill a 1/8” hole in the rear edge of the muffler at the lowest point… then the condensation will be under the car and you won’t see it.
Is your water/coolant level dropping? If not its most likely just condensation. The 1/8" drain holes are always a good idea and will extend the life of the exhaust components. Try parking in the garage 180 deg. out from how your parking now and see if it switches sides. Variations in air flow and temperature gradients can do weird things.