I do a lot of skate skiing on groomed trails. I also do a lot of mountain biking in the warmer months. I do not ride a so-called "fat bike." (This may be very confusing for non-trail-riding Bears who may in the past have heard mountain bikes referred to as "fat tire" bikes. That was a serviceable phrase for years, but now there is a new beast on the scene - mountain bikes with HUGE tires that are more like the size of the rubber on a big motorcycle. These are being called "fat bikes," and I believe are what is under discussion here. The size of the tires means that they can be ridden effectively on packed but not rock-hard snow, where the more old-fashioned mountain bikes would get bogged down.)
As a skier I would not necessarily mind sharing the trails with cyclists if they behaved in a way that showed they understood the issues facing skiers (sort of like they need to understand the issues facing people on horseback or hikers or whatever). However, I might not be excited about what the tire tracks would do to the snow surface. I guess I don't have enough first-hand experience with that to know. It would be totally mandatory for them to stay off the tracks set for classic skiers, because cyclists would obliterate them in one pass if they rode on top of them. It will be interesting to see what the pay-for-use nordic centers have to say. I would expect them to try to accommodate riders, since they represent more tickets and season passes to sell in a short season.
What I do know is that when there is no snow on the ground trails that have the right pitch and width to make for good skate skiing are an utter snooze on a (non-fat) mountain bike. Therefore I have to stretch my imagination a bit to understand why there would be such a big demand to ride on them in the winter with fat bikes. Personally, if there's snow I want to be skiing, so I don't get it.