The answer to your question totally depends where you are and whether you're talking about agriculture or nature. In either case, plants care about moisture present in the soil vs moisture loss through transpiration. This is an American perspective, by the way.

Let's talk agriculture first. In some places, agriculture is rain-fed, for example the eastern US west of approximately the hundredth meridian (e.g., central Nebraska) and also the Pacific coast west of the Cascades and Sierra Nevada. In those places, rivers flowing by don't have much effect on soil moisture except at low elevations near the river--it's all about rain falling on the plants themselves. It is worth mentioning that in a lot of places, impermeable soil means that the ground can only soak up so much rain at a time. So if you get 3 inches of rain in, say, Virginia, you might store 1 inch in the soil and have 2 inches run off into streams, where it's not useful to farms (but may be useful--or harmful--to aquatic life downstream). This totally depends on the soil; sandy soils will soak up water whereas clay soils tend to be especially impermeable, and of course bedrock, rooftops, and paved surfaces don't absorb water. Counterintuitively, soil can be more impermeable when dry, so a heavy rain after a long drought may mean lots of runoff and not much water infiltration.

Natural plants are similar to rain-fed agriculture in that water in streams is not so useful to them, except in the riparian zone.

In other places (the dry western US), modern agriculture requires irrigation. Irrigation can come from rivers, renewable aquifers, or fossil aquifers (e.g., the Ogallala). For river irrigation, the Colorado river needs to be mentioned on its own because of the huge amount of land it irrigates and the huge amount of storage it has. In the Colorado river basin, most of the precipitation falls in mountains near the system's headwaters in Colorado and Wyoming, runs off and is stored in reservoirs, mainly in Lake Powell and Lake Mead (which have enough capacity to store a couple years of river flow), and is distributed to farms in dry places that would not support agriculture otherwise (e.g., California's Imperial Valley), plus lawns, golf courses, etc. Very little is allowed to reach the Gulf of California, a practice that devastated a formerly thriving ecosystem at the Colorado Delta. When we talk about drought on the Colorado river, we mean that less precipitation is falling than usual, meaning that reservoirs are low and still falling. Recovery from drought would mean more precipitation falling near the headwaters; agriculture in the Colorado basin is all about irrigation; given adequate irrigation, the dry climate for the farms can actually be a benefit (more sun, so more productive). Given the hole we are now in and the poor prospects for precipitation to increase in the future, we are probably going to have to adjust our expectations for how much water can actually be drawn from the river.

Other western river systems have parallels to the Colorado but are less extreme. In Idaho's Snake River Plain, for example, agriculture is mainly irrigated by the Snake River, tributaries to the Snake like the Boise and Payette, and renewable groundwater. All of these are mainly fed by snowmelt from the mountains. The region does get some snow and rain through the spring, and that does add moisture to the soil and help delay and reduce irrigation needs; however, it's just a fact of life in the area that the soil dries every single summer and you can't grow crops in the dry season without irrigation water. Reservoir storage (including year-to-year carryover) is significant, but smaller than on the Colorado (less than one year's worth of flow can be stored); the ability to benefit from water stored into the summer as seasonal snow is essential. In this area, recovery from drought mainly means lots of snow.

For fossil aquifers, like the Ogallala on the Great Plains, the aquifer will not be replenished on human timescales, so we should think of it just like we think of a mine: sustainable use is impossible, and once used up it's gone. In dry years, more irrigation water is needed, but the concept of drought does not apply to the aquifer itself.