The problem here is that headlines have space restrictions so they can't always be as verbose as they should be to construct a perfectly grammatical and understandable sentence. The point of a headline is just to give you the basic idea of what the article is about, and the article itself will explain things more precisely.
In this particular case, the first half of the sentence is simple enough. It is saying that one out of every eight residents of Denver is depressed. "One" is singular; the sentence is saying that the one resident (out of the eight residents) is depressed. If the study would have shown that two out of eight residents are depressed then the headline would have said "Two in eight Denver residents are clinically depressed". This is because once there are two residents being discussed, it is now plural.
The second half of the sentence is a little more complicated, because it's not entirely clear who "most" is referring to. Just going by the sentence structure, it could be referring to "Denver residents". If that was the case then "are" is appropriate because "most Denver residents" is plural. On the other hand, when we take into account the content of the headline, it seems more likely that "most" is referring to the people who are depressed (because why would anyone expect non-depressed people to get treatment for depression?). Now it gets a little confusing because the "one" from the first half of the sentence has changed from referring to a single person to referring to multiple people. This is because the first half of the sentence was simply stating that one out of every eight people is depressed, but in reality there are many more than eight people in Denver. If there are eighty people then ten of them are depressed; if there are eight hundred people then one hundred of them are depressed. The discussion has switched from the one person out of eight people to the sum of all the one persons in Denver. Thus, the "most" that are not getting treatment consists of many people (the sum of all the ones from the first half of the sentence, i.e. 12.5 percent of Denver's population). Since it is now plural, "are" is appropriate. Had there not been space constraints on the headline, the sentence might have been written more clearly as:
One in eight Denver residents is clinically depressed, and most of the depressed residents are not getting treatment.
In fact, in the actual article it turns out that "most" wasn't even referring to most of the depressed residents in Denver either. Rather it was referring to most depressed residents in the entire state of Colorodo:
To compound all that news, almost 70 percent of Coloradans who are depressed are not getting treatment.
Now we can see even more clearly that the headline was imprecisely written in order to convey as much information in as few words as possible. In any case, "70 percent of Coloradans" is also plural and would thus get the word "are" anyway.