Those trees are probably Thuja plicata, also known as Western Red Cedar or Arbor Vitae, but could also be junipers of some kind or a Western US species that I'm not familiar with. In any case, two trees of the same species planted that close together should be equally affected by watering issues, because their roots are definitely overlapping each other in the soil, and by many other issues because of their proximity. Additionally, a recent drought would not show damage this quickly, as it can take weeks to months for an evergreen to show foliar issues. I would rule out watering then and look for other causes.
I think you could look at (in order of likelihood):
- Wounds to the roots or trunk
- Chemical damage
- Nitrogen deficiency
- Insect or disease
Has there been any trenching or other construction near the tree? Of so, this could be the cause of the yellowing, although I'd expect the damage to be a bit more localized. If not, then you can rule this out.
This could be drift from an application of something like RoundUp to an area to the right of the photo, but then I'd expect the damage to be only to that side of the tree and also perhaps affect the same side of the other tree. Alternatively then, a chemical could have been deposited on the lawn on that side of the property and taken up into the tree via the roots. Perhaps a spill of some kind? Perhaps someone dropped way too much fertilizer on the right side of the lawn? This may have occurred many feet from the trunk of the tree (if arbs, then the tree roots spread out quite a ways from the tree).
I think that this is unlikely, as it would most likely affect both of the trees. Its symptoms are a yellowing of leaves through the tree, which does match what you're seeing, however, so it may be worth trying a soil test in the root zone of the affected tree (say, within 10-15 feet of the trunk). Do NOT treat for this without a soil test confirming low nitrogen, though!
Insect or Disease
The best person to identify whether your trees are infested with anything is a certified arborist, but the fact that one tree is perfectly healthy and the other is in trouble kind of suggests that these are not a factor in the problem (both trees should be equally affected).
Ideas for these solutions come from the Ortho Problem Solver reference book.
One reference for you could be your state's Extension service. Most states' Extensions have websites (usually available from the state university with the best horticulture program). These have information that discuss local problems with trees, shrubs, and other plants and could perhaps point you in other, more localized, directions.