I googled a bit: almost the entire Kamniška Bistrica is one big parcel (cadastral municipality 1891 ŽUPANJE NJIVE, parcel number 873/174 Area 27,321,424 m2). As I quickly looked, the parcel extends into all the valleys and dead ends around there (Korošica, Kurja dolina, Konec, Jermanca, Bela,...), only here and there are individual "islands" that don't fit into it.
The parcel is owned by about 100 to 150 natural persons (land registry entry not yet completed, I assume it could be members of the Bourgeois Corporation, some agricultural community from Kamnik), two smaller shares are owned by RKC institution. The road is not excluded from the parcel (like the bed of Kamniška Bistrica), but the Municipality of Kamnik has a servitude entered on the parcel, also for transport with various vehicles. I'm not clear if this servitude applies generally to everyone or if someone can restrict it, e.g. with traffic signs. So it's private land.
Police generally shouldn't enforce on private land. If a neighbor parks in my driveway, police won't come to fine him, it's a matter of civil proceedings!! Therefore, police interventions deserve at least explanation. Also very unusual is the story with the sign that sometimes is there, sometimes not, because the municipality removed it.
Overall, it seems to me like some political PR tussle, on the backs of fined visitors who are just collateral damage or even forcibly engaged as media to motivate the public. Recently there was news of another attempt to establish Kamniške Alpe national park, which the Bourgeois Corporation opposes. They think they would have too little influence on their parcels. The scenario could of course be different...
Anyway, probably control over a huge estate that is also very valuable is at play.