Actually you declare the war, and then you may engage their members in combat freely (meaning with no alignment losses) and you may loot the enemy members' personnal claims that are outside of a city protected radius like it's judgement hour.
if you want to lower an enemy city's monument level, you declare a battle. Win battles enough time and monument will be back to level 0. Then you may declare a siege, and when it enters the actual sieged state, the city itself is lootable like judgement hour until the monument has been taken out or if the defenders hold out for long enough. Note that battle will be instanced, so no one that hasn't been "invited" will be there. However, sieges will not, so basically anyone can take the opportunity to steal valuable stuff in the city.
That's how it'll work
Of course, if a country has declared oath of allegiance to another country (thus creating / expanding a kingdom), you'll have to declare war to the whole kingdom, not just the country you're targeting.