How do the Drought and Snow Warning abilities manifest in Pokemon? Do those Pokemon carry around their own personal weather cloud or are they capable of instigating the weather effects as they please?

Loosely speaking, the latter, yes. Basically, these pokémon emit powerful auras (not literally Aura but rather an aura of power) that influence the immediate atmosphere. So if a pokémon with Drought walked into a room, the air would turn hot and dry. Some more powerful pokémon can even manifest fireballs that form miniature suns. If released outside, then these auras will automatically trigger Sunny Day or Sunny Day-like effects. Conversely, Snow Warning chills the air to the point where the moisture in the room forms its own weather system, complete with hail. While one might assume you can counter such an ability with a dehumidifier, it should be noted that the secondary effect of Snow Warning is to release enough moisture in the air for its primary effect to occur.

In short … pokémon essentially conjure weather effects wherever they go, by manipulating the atmosphere around them with their inherent elemental abilities.