I have a Lilac bush that flowers but only has a half a side of green leaves.
Certified GKH Gardening Expert
I suspect that you have had a cold spring and that the side of the lilac that has leaves if facing a wall or structure which is creating a bit of a micro climate. The side outside the microclimate is slow in leaving out because of the cold, but the other side is getting heat from the wall/structure and this is making it warm enough to leaf out.
The other possibility is that the tree has a fungus that has not yet reached the other side. If you want to be on the safe side, treat the shrub with a fungicide.