> But how does the field look like on the other side of the plate? Is it zero?


No there are charges induced on both the surfaces equal in magnitude but opposite sign so as to cancel the effect of electric field of the charge q(your point charge) going through the sheet.

Now because Negative charge is attracted towards positive charge the side of sheet facing the charge q will accumulate a charge -q with the centre of charge at the point you calculate using mirror symmetry and the positive induced charge on the other side isolated and uniformly distributed.