As is also true in mammalian lung, the endothelium of the capillary is extremely tenuous, and so is the epithelium of the adjacent air capillaries, so as to provide only the minimum barrier to gas exchange. Any given blood capillary is in continuous contact with several air capillaries (AC) (and vice versa) through the whole thickness of the wall of the air vesicle. The air vesicles of this parabronchus are separated from the adjacent ones by septum of connective tissue (S).
In a high power image like the next one you can better make out the relationship of the two types of capillary. The air capillaries are recurrent loops; they begin in one part of the air vesicle and loop back to it. Air moving through the lung moves through these thin epithelial walled vessels in one direction.
The blood capillaries move blood in the opposite direction. Thus as the blood absorbs oxygen across the barrier of the two epithelial layers (the air capillary wall and the blood capillary endothelium); it's always moving into a stream of air in which the oxygen content is at maximum levels. This counter-current flow is responsible for a good part of the efficiency of avian lungs. The close proximity of one blood vessel to several air vessels and vice versa also promotes rapid and complete saturation of the blood with oxygen.
Unlike the tidal in and out ventilation of mammalian lungs, in which a part of the volume is "stale" air, and unlike the mammalian structure with a lot of blind alleys and dead space, the avian lung is a continuous flow system. Air comes in and goes out in a more or less steady stream, propelled by the bellows action of the air sacs and constantly coursing over the minute blood vessels. Birds need a system like this to provide for their very high demands for oxygenation of the massive flight muscles.
Avian lung; H&E stain, paraffin sections, 200x and 400x