Rio Grande del Norte National Monument is BLM land so it is all dispersed camping. You will need to make sure you aren't on someone's property or nearby any roads. There are plenty of turn offs.
This isn't a "get away" more like a stopover spot. I stopped here on my way back North to Colorado to see the bridge over the Rio Grande and so I wasn't so high up in elevation that I would freeze.
But if you can find a quiet place it is a beautiful dispersed spot where there is civilization nearby and toilets at local businesses and picnic areas that you can use. See map on review.
Be careful roaming at night. The drop into the river is quite deep so offroading isn't just ill advised for keeping the flora and fauna safe…but you as well.
There are picnic ares around the monument too, just not established camping spots.