Graynomad:
RE is active low, on the right-hand chip it's pulled high.
Rob
Aha, thanks for that hint! I thought something might be up with it, with the slash after the name or the "overscore". Learning by doing, I guess 
So I need to connect it to ground on the right side one then.
I found this table in the datasheet:
So on the left side, RE can be disconnected, DE to VCC.
On the right side, RE to GND, DE can be disconnected.
Like this?