jremington:
Just make those pins INPUT pins, and they are effectively OFF.
I just tried this tip and it works! Cant believe the solution would be this easy 
And this is the cheapest solution too... I thought it was not possible in software..
CrossRoads:
Add a 74HC125 to each Arduino. Have one Arduino control the output enables of each chip - only turn on one chip at a time.
This IC looks promising, maybe I try one on a boring day. For now I have my solution but thank you very much for mentioning the IC.
Paul__B:
So what is this other device and what is the purpose of the whole arrangement?As has been pointed out, it is perfectly easy to set the Arduino pins as INPUT, but if the other device cannot do this you simply use a 74HC4066 as a "relay". There is some concern regarding how these devices will be powered on and off.
Sorry but I really dont want to waste your time and energie with my XY Problem.
But thank you for repeating what other people already mentioned.
david_2018:
Would probably be better, especially in the case of two arduino's, to just have one arduino control the display, and the second communicate with it when something needs to be displayed.
MarkT:
But why? This isn't how displays are driven. The display is driven by some controller, and whatever wants
to talk to the display talks to the controller, not to the display - its a black box effectively.
I will think about this kind of communication order.. Maybe its better than my plan.
Thanks everyone for the fast answers and hinds!
Phil