Hello Finn Hyttel,
Sorry for late reply to this,
In my suggestion, it is possible to do this but it cannot run 2 forum in 1 site, just make a technical categories permission setting which group can view the categories, which can't view that categories in 1 forum.
You need to set the categories permission from your backend example :
First (Public Categories)-> Create a categories can allow all the user view, reply and create new discussion set from backend > Component > Easydiscuss > Categories > Permission tab
Who can select this category when creating discussion -> Registered and Guest group
Who can view discussion in this category -> Registered and Guest group
Who can reply discussion in this category -> Registered and Guest group
Who can view replies in this category -> Registered and Guest group
Who can moderate this category -> SuperUser group
See the video link >
http://screencast.com/t/BMPT8tnC9aVp
Second (Private Categories) -> Create a categories and set the permission :
Who can select this category when creating discussion -> Administrator group
Who can view discussion in this category -> Administrator group
Who can reply discussion in this category -> Administrator group
Who can view replies in this category -> Administrator group
Who can moderate this category -> SuperUser group
Hope you can understand my explanation.
Best Regards
Arlex