Your direction was OK. export DISPLAY is what you want. But it is not enough.
On the target, you need to type
xhost +from.where.the.windows.are.coming.com
It gives the X server the permission to allow remote windows from this computer.
Beware, it is not really secure! A possible attacker could not only windows shown by you, but even control your mouse/keyboard. But for simple solutions, or if you can trust the remote machine and the network between you, it may be ok.
If not, there is an advanced authorization, based on preshared keys. It is named xauth. Google for xauth.
The Xorg server has an option to disable the remote windows, and there are distributions, (f.e. ubuntu!) who turn this option by default on. You can test it - if you can telnet to the tcp port 6000, it is allowed.